Early Alfonsine Astronomy in Paris: The Tables ofJohn Vimond (1320)

José Chabás and Bemard R.. Goldstein

lt has beco clear for many years lhat medieval European astronomy in Latin \Vas heavily dependent 00 sources from the Iberian península, primarily in Arable, bUI also in Hebrew, Castilian, and Catalan. The Castilian Alfonsine Tables, compiled by Judah ben Moses ha.cohen and Isaac ben Sid under the patronage of Alfonso X (d. 1284), weTe ao importanl vehicle for the transmission of this body of knowledge lO astronomers north of the Pyrenees, bUI the delails of Ihis transmission remain elusive, in part because only the canaos lO these tables survive (sec Chabás and Goldstein 2003a). In Ihis paper we build 00 OUT preliminary studies of a figure who previously had barely beco mentioned in the receot literature 00 medieval astronorny (Chabás and Goldstein 2oo3a, pp. 267­ 277, and 2003b). John Virnond was active in Paris ca. 1320 and, as we shall see, his tables have much in common with Ihe Parisian Alfonsine Tables (produced by a group in Paris, notably John of Murs and 10hn of Ligneres), bu! differ from them in many significant ways. As far as we can tell, there is no evidence for any interaction between Vimond and his better known Parisian contemporaries and in our view the best hypothesis is that they al1 depended on Castilian sources. As a result of our analysis, we are persuaded that Vimond's tables are an intelligent reworking of previous astronomical material in the Iberian peninsula to a greater extent than is the case for the Toledan Tables (compiled in Toledo about 2 centuries before the Castilian Alfonsine Tables). It is most likely that Vimond's principal source was the Castilian version of the Alfonsine Tables. Paris, Bibliotheque nationale de France, MS lal. 7286C is a 14th· century manuscript containing an unusual sel of tables (ff. Ir-8v) as well

Suhayl4 (2004) 208 J, Chabás and B. R. Goldstein

as the canaos and tables of 1322 by John of Ligneres (ff. 9r~58r). In a brief text al the end of the first sel of tables they are attributed lo John Vimond (Iohannes Vimundus), an astronomcr who compiled them "for the use ofstudents at the University of París" (f. Bv):

El ill hoc termillalur opus /ohannis Vimundi baiocellsis dyocesis de disposicionibus plalletarum el ste//arum fixarum; el cum istis sequitur de hiis que per ipsum ordinantur ad cOllversionem temporum verorum el equalium sociatorum, el de disposicionibus eclipsalibus solis el lime sibi pertil1entibus, el de aliis disposiciollibus ipsorum el aliorum corporum celestium, ad u/nitatem seolarium universitatis parisiensis el offlnium a/ionllll.

Here ends the \Vork by John Vimond of the diocese of Bayeux on the dispositions of the planets and Ihe fixed stars; (... ) and on the dispositions of solar and lunar eclipses and [other syzygies] corresponding to them, and on the orher dispositions ofthese and other celestial bodies, for the use of students at rhe University of Paris and a11 others.

The complete set of Vimond's lables are uniquely extant in this manuscript, and no canons for them have been identified. They are a coherent set of tables with a11 the elements needed to compute the positions of the celestial bodies, much in the tradition of the Arabic zijes and their derivatives. The exact date of composition of Vimond's tables is not given in the text, but they were probably produced shortly before 1320. In the paragraph preceding his tables, Vimond tells us that they were compiled for Paris with 1320 as epoch (f. Ir: see below) and this date is confinned by recomputation. These tables also include a calendar with the dates ofsyzygies: this strongly suggests that they were constructed prior to the year of the calendar because lhe astronomical information would no longer be of any use after the year had passed. However, the calendar poses special problems which will be discussed below. Vimond's only other known work is a short treatise on the constrUClion of an aSITonomical instrument, extant in Erfurt, MS CA 2° 377 (fr. 21r-22r), beginning Planicelium vero componitur ex eis que su"t diversorum operum.", and ending Explicit tractaius johannis vimundi... in a manuscript containing vatious works by other Parisian astronomers Early Alfonsine ASlrollolllY in Paris: The Tables o[Johll VimOlld (J 320) 209

such as Joho of Murs aod Joho of Ligneres (Thorndike and Kibre 1963, coL 1050; Saby 1987, pp. 471, 474). John Vimood aod his works were seldom meotiooed by his cootemporaries. However, io Vaticao, Biblioteca Apostolica, MS OUob. lat. 1826, we are told that Joho of Spira (14th ceotury), the author of a cornmentary on Joho of Ligoeres caoons (Thorndike aod Kibre 1963, col. 204), composed his own caooos to several oC Vimood's tables (for a descriptioo ofthis maouscripl, especial1y fe. 148-153, see F. S. Pederseo 2002, p. 177). This manuscript ineludes a text that begios on e. 148ra ascribed lo a certain M. J. c., Canon tabulae sequentis quae inlilulatur tabula lIIotus diversi solis et lunae in una hora el semidiametrorum secllllllum labulas Alfonsi, at the eod ofwhich Joho Vimood is mentioned. 00 the other hand, Vimond is nol menlioned in Madrid, Biblioteca Nacional, MS 4238, a manuscript containing a few tables that can be altribuled to him, as well as a copy of Ihe Parisian Alfonsine Tables compuled for Morella (io the provioee of Valencia) for the years 1396 and 1400 (Chabás 2000). As far as we can tell, Joho of Murs aod Joho of Ligneres do not refer to Vimond at a11 in any of their numerous works, but it seems implausible that they did not know him or his work which was addressed to !he sludents at the University oC Pans. Indeed, there were not so many competent astronomers working in Paris around 1320 aod both Vimood and Murs carne from the same regian, Nannaody, from places about 70 km apart, Bayeux and Lisieux, respectively. We wauld expeet Vimond to be wen known aod frequeot1y cited by practitiooers of astraoomy, for he is oamed as one of the outstanding astronomcrs of his time by Simon de Phares in his Recueil des plus celebres aSlrologues (1494--1498), a chrooologically ordered list with comments, edited by Boudet (1997-1999, 1:467). In faet, Vimond is mentioned before John oC Ligneres, John of Saxony, Joho of Janua, aod Joho ofMurs:

Maistre Jehan Vymond fut a Paris, }¡omme moult singulier el granl aSlrologien, lequel eut en ce temps grant cours pour la sciellce des estoilles. Entre ses euvres, jist ulle verifficacioll de la conjU1/Clion des lu[mi]naires, al/ssi des eclipses el eSlOil!es fixes pour plusieurs ans. Ceslui predist les grans vellS qui Juren! en son lemps el jiSI plusiellrs beaulx jugemens, donl il acquist gran/ loz el renommee en Frallce el fw moltlt devos! en Noslre Seiglleur. 210 1. Chabás and B. R. Goldstein

Master John Vimond ¡ived in Patis, a most singular man and a great astrologer, who had at that time much prestige because oC (his knowledge oi) the science of the stars. Among his work:s is a verification afthe conjunction ofthc iuminaries, as we1] as eclipses and the fixcd stars, for many years. He predicted the great winds which took place in his time and marle many fine judgments for which he acquired great praise and renown in France and he was most devoted to our Lord.

The "verifficacion de la conjunction des lu[mi]naires" refers to Vimond's tables. These tables are arranged very differently from those of his Parisian contemporaries and are based, in part, on parameters 1hat probably. carne [rom the Castilian Alfonsine Tables or a tradition c10sely associated with them. Of special interest is the proper molion of the solar and planetary apogees, a feature previously unknown in medieval tables produced outside Spain and North Africa. We are convinced that Vimond's tables provide an indication of the arrival in Paris of new astronomical material coming from Castile, in the sense that they propose new approaches to replace those based on the Toledan Tables and developed al the end of the 13th century by astronomcrs working in Paris such as Peter Nightingale, Geoffreoy ofMeaux, and WiIliam ofSt.-Cloud. Furiher, we believe that Vimond's tables are prior to, and independent of, the tabular work developed in the early 14th century, which we call the Parisian Alfonsine Tables, by the group of Parisian astronomers that included John of Murs and John of Ligneres, which were also based on Castilian sources. Vimond's tables and the Parisian Alfonsine Tables have many parameters in conunon both for mean motions and equations. In principie, it is possible that one set oftables depended on the other, but the differences between them suggest to us that it is far more likely that they depended on a cornmon source. Moreover, ifVimond composed his tables prior to 1320, he did so before any datable text of the Parisian Alfonsine Tables. A description and analysis of Vimond's tables follow.

f. Ir The first numerical infonnation given in this set of tables is the "radix for mean conjunctions of the Sun and the Moon": 13;54,54d. In modem tenninology, the initial time for a set oftables is called its "epoch" whereas its "radices" are the positions of the Sun, Moon, planets, etc., at Ear/y A1foflSi,.~ Astronomy in Pori$: T1r~ Tobles o[Joh,. Yimond (JJ20) 211

that time. The medieval convention, however, is to use "radix" for both the time and the position. \Ve are convinced that the author refers to the time, in Pans, orthe mean conjunction on March lO, 1320. The year and the place are mentioned by Vimond himselr in a short paragraph following the numerical value of the radix (f. 1r):

Et est ime/ligendum quod ista radix mediarum coniuncciOltlmt sil immediate post 19 secunda diei que consiSlUtlt immediate post (. ..) lucis beati Mathie eomposite procedendo ab onu solis usque ad oecasum seilieet anno domini nostri ¡hesu Christi 1320 secundum numeracionem annorom romanorom qui ineipiunt ex ¡melO diei eirconeisionis domit¡i nostri ¡hesu Christi et existentis ad longitudinem civitalis Parisills que distat a medio mundi per 49 g et 30 min ila quod illa eivitas est in parte oecideltlali et etiam distat ab illo medio per 8 min et l5s diei equalis.

Note that this radix for the mean conjunctions comes immediately afier 19 seconds of a day that fall immediately after the (...) [space for one word; iIlegible] (day)light of Saint Matthew, proceeding from sunrise to sunset, namely, in the year of our Lord lesus Christ 1320 according lo the count ofRoman years which start from the beginning ofthe Day of lhe Circumcision of our Lord lesus, for the longitude of the city of Pans which is distanl from the middle of the world by 49 degrees and 30 minutes because thal city is in the westem direction and distant from that middle by 8 minutes and 15 seconds ofan equinoctial day.

Madrid, Biblioteca Nacional, MS 4238, f. 66v, has a short text which is very similar to the paragraph on f. Ir in Paris, Bibliothequc nationale de France, MS laL 7286C:

Radix coniultccionum: dies 13 m 54 2" 54 Radix opposicionum: dies 14 m 45 2" 55 Hec radix est post 19 2" diei que sunt post meridiem diei Malhie onno 1320°secundum romanos. Nota quod annus in meridie diei mathie 25 dies bisexti erit semper ultimus dies 212 J. Chab!s and B. R. Goldstcin

anni. lsre tabule radicum sunt lacte Parisius ad meridiem cuius cenith distal ab equinoccia/i 49 g 30 m vel 8 m J5 2" diei.

The numerical datum, 0;0, t 9d (= 0;7,36h), represents the equation oftime for that day.ln the Madrid version,lhe "radjx orlhe opposition", 14;45,55d, is half the length of a mean synodic month which is about 29;3 I,50d, and Ihis is the entry for the first opposüion in Vimond's calendar (see below). It is clear Ihat, according to the version ofthis text in the Paris manuscript, the civil day in the calendar ¡neludes the period of daylight, Ihal ¡s, the lime from sunrise lo sunset, in contrasl to the astronomical day tha! goes from noon to the following noon. Both values given for ¡he longitude of Paris from Arin, caBed "lhe middle of the world", are equivalent. Arin, a corruption of Ujjain (a city in India), was thought to be halfway between the eastem and westem Iimits of lhe world (Neugebauer 1962, p. 11, n. 2). The distance from Arin to Toledo was taken to be 61;30° and, since Paris was generally said to be 0;48h or 12° to lhc east of Toledo, its longitude from Arin is 49;30°, as in the passage above (Millás 1943~1950, p. 49; Kremer and Dobrzycki 1998, p. 194; and F. S. Pedersen 2002, p. 431). Moreover, when a day is taken to be 360°, it follows that 49;30° corresponds to 0;8, 15d, for 49;30°/360° = 0;8,15. The expression in the Madrid manuscript, "Parisius ad meridiem cuius cenith distal ab equinocciali" is a corrupt version ofthe better reading in the Paris manuscript, for it would imply that 49;30° is the latitude of Paris, but then its equivalence to 0;8,1 Sd would become meaningless. We also note that, according to this text, Vimond's tables were computed for Pans whereas in the early 1320s other Parisian astronomers who recast the Alfonsine Tables computed them for Toledo, as is the case for the tables with epoch 1321 by John of Murs (see, e.g., Lisbon, Biblioteca de Ajuda, MS 52-XJ]­ 35). According to our computations based on the Parisian Alfonsine Tables, the mean conjunction on March 10, 1320 took place in Toledo a1 9; 1Oh, civil time (Le., counting from midnight) which, with a correction of 0;48h, is 9;58h in Paris (civil time), that is, 2;2h before noon. Thus, the radix for the tables on f. Ir (as well as the radices for the planetary tables, as wil1 be seen later) is the time of the first mean conjunction in March 1320 (March lO, 1320, at 9;58 a.m., Paris, or March 9, 1320, 21;58h, Pans, counting from noon). lndeed, the sexagesimal part of the "radix" (0;54,54d) is exactly the sum of 12h and 9;58h. The integer part of the "radix", as wi11 be explained later in reference to the annual calendar Early Alfonsine Aslrollomy in Paris: TIle Tables 01Jolm V¡mond (1320) 213

presented on this same folio, is counted from the epoch of the calendar, almost 14 days before the mean conjuncrion ofMarch 10, 1320, that is, February 25, 1320 or February 24b, 1320, where 24b represents the second day caBed Februal)' 24 in a leap year (such that the last day of February is always day 28 bolh in ordinary and leap years).

f. Ir Table 1: mean conjunctions

The entries in this table give the instant of the first mean conjunction after a certain number of years. We are given entries for 1,2, 3, and 4 years; for multiples of 4 years up to 76 (= 19·4) years; and for 152,304,608,1216, and 2432 years. The entries represent ¡he excess of days after an integer number, n, of synodic months have elapsed (where n = 13 for year 1, ..., and 30,081 for year 2432). Madrid, Biblioteca Nacional, MS 4238, f. 66v, reproduces this table except for the last row for 2432 ycars, which is missing.

Table 1: mean conjunctions (f. Ir)

[years] [excess] [years] [excess] [ycars] [excess] d d d

1 18;53,52 28 20; 6,54 64 12;13,41 2 8;15,53 32 6; 6,50 68 27;45,27 3 27; 9,45 36 21;38,37 72 13;45,23 4 15;31,46 40 7;38,33 76 29;17,10 8 1;31,43 44 23;10,19 152 29; 2,29 12 17; 3,29 48 9;10,15 304 28;33, 8 16 3; 3,25 52 24;42, 2 608 27;34,26 20 18;35,12 56 10;41,58 1216 25;47, 1 24 4;35, 8 60 26; 13,44 2432 21;43,12

The value for the mean synodic month derived from year 2432 is 29;31,50,7,44,35d ± O;O,O,O,O,4d. Thus, ror ye,r 1; 13 . 29;31 ,50,7,44,35d - 365d = 18;53,52d, in agreement with the tabulated valuc. In the Parisian Alfonsine Tables, the mean synodic month is 29;3 1,50,7,37,27,8,25d: this 214 J. Chabás and B. R. Goldstein value is found, for example, in Lisbon, Biblioteca de Ajuda, MS 52·Xll­ 35, r. 16v, containing the tables for epoch 1321 by John of Murs. So Vimond's parameter is very similar lO, but 001 identical with, the parameter in the Parisiao Alfonsine Tables.

Vimond Parisiao Ale T.

Mean synodic month 29;31,50,7,44,35d 29;31,50,7,37,27,8,25d

f. Ir Table 2: annual calendar with syzygies

This annual calendar begins on the day of Saínt Matthew (February 24) and Iists the dates associated with several saints, as weU as the dates and times of 25 consecutive mean syzygies. The practice of adding the extra day in a leap year aftcr Feb. 24 goes back lo the Roman calendar as revised by Julius Caesar, when the additional day folJowed Feb. 24 and was called bis-sextus ante calendas mar/ios (the sixth day befare the calends of March). In a ¡eap year February lasted 29 ~ays. bul the lasl day was numbered "28". for the 24th was assigned lo two conseculive days. This is what is intended in Vimond's calendar where lhe year begins on that very day. We know of no other calendar in the late 13th ceo~ry or early 14th century beginning 00 Feb. 24; in particular, the calendars composed by Gcoffreoy of Meaux and William of St.·Cloud do not begin on that day (Chabás and Goldstein 2oo3a, pp. 245-247). It is worth noting that Vimond's calendar which lists mean syzygies together with sainlS' days is in the tradition of these two astronomers who were active in Paris shortly before him: they displayed planetary data in calendars and depended on the Toledan Tables for their computations. We also note that the feast of St. Matthew is mentioned in the canons to the Parisian Alfonsine Tables by John ofSaxony as the last day in a leap year (see Poulle 1984, p. 36, line 41). Vimond offers no explanation for basing his calendar and his tables 00 syzygies; we can only conjecture that he was being faithful to sorne unknown source. In Table 2, columns I and 2 have no heading, but column 3 has the heading "days, minutes, and seconds·'. In the manuscript the name of the month is usually given in col. 2, and occasionally in col. 1 which has about 90 entries such as: Annunciarjo Domjnj, Dyonisius, Lucas Evangelista, Innocemes. etc. The syzygies are numbered from 1 to 25, and they are transcribed below. The numhers in column 3 are integers when a saint's day is meant and indicate the number ofdays that elapsed since the epoch Early Alfollsine As/rOllomy in Paris: Tire Tables o[Jo/m Yimolld (1320) 215

(day O) ofthe calendar, that is, February 25, 1320 (Julian) or what we have called February 24b, 1320. Vimond seems lo use here civil days (from midnight to midnight) rather than astronomical days (from noon to noon), which makes sense in a calendar. When a conjunction or an opposition is indicated, we would expect the number in column 3 to refer to the accumulated lime from the radix (the conjunction on March 10, 1320) in multiples ofhalf a mean synodic month, Le. 14;45,55d but, in fact, we are given the accumulated time from the mean syzygy-an opposition­ immediately preceding the radix, which occurred on February 24, 1320. lf this \Vas the author's intention, it is not clear how the user of these tables was to take account ofthe radix given at the beginning of Ihem. Moreovcr, despite Ihe coherence of the arithmetic in this calendar, somelhing is seriously wrong with it, for we find the word oppositio next to March 10, when a conjunction took place, and Ihe word coniu/lccio nexl to March 25, when an opposilion occurred. The same pattem IS followed throughout the calendar. There is an "explanatory" note on f. 1r conceming the calendar, but we were unable to make sense of it.

Table 2: annual calendar with syzygies (f. Ir)

(1) (2) (3)

[Saint's day [date] [time since epoch] I No. syzygy] d

Romanus February 28 4 Perpetua virgo March 7 11 1 Opposition March 10 14;45,55 Gregorius papa 12 16

2 Conjunction March 25 29;31,50

3 Opposition April 9 44;17,45

4 Conjunction April 24 59; 3,40

5 Opposition May8 73;49,35

6 Conjunction May23 88;35,30 216 1. Chabás and B. R. Goldslein

7 Opposition June 7 103;21,25

8 Conjunclion June 22 118; 7,21

9 Opposilion luly 6 132;53,16

10 Conjunction luly 21 147;39,11

11 Opposition August 5 162;25, 6

12 Conjunction August 20 177;11, 1

13 Opposition September 3 191;56,56

14 Conjunction September 18 206;42,51

15 Opposition October 3 221 ;28,46

16 Conj unction October 18 236;14,41

17 Opposition November 2 251; 0,36

18 Conjunction Novembe:r 14 265;46,31

19 Opposilion November 31 280;32,26

20 Conjunction December 16 295;18,21

21 Opposition December 31 310; 4,16 Circoncisio domini filesu Christi ¡"i(ium anlli [January 1] 311

22 Conjul1ction January 14 324;50, II

23 Opposition January 29 339;36, 6

24 Conjunction February 13 354;22, 2 luliana virgo 16 357 Perros ad carhedram 22 363 2S Opposition February 28 369; 7,57 Eariy Alfonsilre ASlrOllomy j'J París: The Tables 01Jolm VíllJolld (/320) 217

Year 1324 might be considered as an altemative date for the calendar for, according to computations with the Parisian Alfonsine Tables, a mean opposition occurred 00 March 10 (counting from noon) or on March 11 (counting from midnight). This would conform with the character of the syzygy mentioned in the calendar, and the computations associated with this date yield results that are quite clase to (but not exactly the same as) the informarion given in the texl. lndeed, in our preliminary discussion of these tables, this near agreement mislead us to think that 1324 was the radix of Vimond's tables (Chabás and Goldstein 2003a, p. 270). However, as indicated previously, year 1320 is specifically mentioned, and it fits much better with the radix of mean conjunctions and wíth lhe radices for planetary positions displayed on ff. Iv and 4r.

f. Iv Radices for the argument of solar anomaly, the argument of lunar anomaly (henceforth, solar and lunar anomaly, respectively), the solar apogee, and the lunar ascending node:

Solar anomaly 8s 26;14,33° Lunar anomaly Is 3; 6,14° Solar apogee 2,29;56,15° Ascending node lOs 13;14,43°

Note the use of signs of 30°, a characteristic of a11 tables in this sel. A short text belo\V these parameters explains that the radices for the motion of the solar apogee and the ascending node are counted from the beginning of Aries 00 the 9th sphere, indicatiog that tropical coordinates are used here. These radices were calculated for March 10, 1320, at the time of the mean conjunction of the Sun and the Moon. According to the Parisian Alfonsine Tables the solar apogee for March 10, 1320 is 89;23,50°, a value which differs by about half a degree froro the entry in the text. 80th values in turn differ from the solar apogee for 1320 in the tables for 1322 by John of Ligneres (89;24,22°) that is found by adding two values given on f. 9v ofthis same manuscript the solar apogee (81;7,15,39°) and the motion of ¡he 8th sphere at that time (8; 17,6,48°). The same result, 89;24,22°, can also be found in another copy of John of Lignercs's tables, Erfurt, MS CA Q 362, f. 21 ra. For the rest ofthe radices, recomputation with the Parisian Alfonsine Tables for the epoch, March 9, 1320, at 21;lOh in Toledo (counting from noon), yields results which are very close 10 the values in the text, especially for the Moon: 218 1. Chabás and B. R. Goldstein

Vímond Parisiao Alf. T.

Solar anomaly 266; 14,33° 266;47, 0° Lunar anomaly 33; 6,14° 33; 6,28° Solar apogee 89;56, J50 89;23,500 Ascending node 313;14,43° 312;54,39°

The solar longitude is the sum of the solar anomaly and the solar apogee:

Vimond Parisiao Alf. T.

Solar longitude 356;10,48° 356; 10,50° and again the agreement is ve!)' good. Since this is the time of a mean conjunction, the mean lunar longitude will be egual to the mean solar longitude. According to the Parisiao Alfonsine Tables, the mean lunar longitude at {his epoch was 356; 11,3°, ¡.e., it differed from the mean solar longitude by only 0;0, 13° (note that the Mooo travels this distance in about 20 seconds aftime which is below the accuracy of 1 minute for the time of mean conjunction). Hence the absence oí a radix for lunar mean motion simply renects the fact for the epoch of Vimond's tables the mean longitude oí the Moon is the same as the mean longituge ofthe Sun. The agreement for the radix of lunar anomaly lo the minute is particularly impressive since the motion in lunar anomaly is about O;30o /h. Lunar anomaly is not subject lO precession and it is independent of solar motion. (We use the term precession for a constant motion of the eighth sphere, and Irepidation for a variable motion of the eighth sphere.) So, even though Vimond and the authors of the Parisian version of the Alfonsine Tables differ on matters oí definition and marle slight changes in mean motions, it is unlikely that either of thero would change the motion in anomaly significantly from what it had been in their common source. Early Alfonsille AstrollOmy in Paris: The TablesofJohn Vimond (1320) 219

f. lv Table 3: yearly radices

This table displays the radices for the solar anoma1y, the lunar anomaly, the solar apogee, and the lunar ascending node for t, 2, 3, and 4 years; for multiples of4 years up lO 76 years; and then for 152, 304, 608, 1216, and 2432 years, as in Table l. A selection of the entries is displayed in Table 3.

Table 3: yearly radices (f. 1v)

Year I Year 2 Year 3

S el s e) s (0)

Solar anomaly O 18;22, 3 O 7;37,47 O 25;59,49 Lunar anoma1y 1I 5;37, 8 9 15;25,15 8 21; 2,22 Solar apogee O O; 1,12 O O; 2,18 O O; 3,30 Ascending node 1I 9;43,23 10 20;58,27 lO 0;40,50

Year 60S s (0)

Solar anomaly o 1;24,48 020;6,6 Lunar anomaly I 5;51,58 4 8;22, 2 Solar apogee O O; 9, 7 O I1 ;32,36 Ascending nade 6 25;27,27 4 29;26,44

The entries for year 1 represent the progress made by the Sun, the Moon, the solar apogee, and the lunar node in ayear of 13 mean syzygies of the same kind (henceforth "Iunations") of 29;31,50,7,44,35d. To be 220 J. Chabás and B. R.

sure, the difference between the solar anomalies for year 2 and year 1 is 349;15,44°, meaning that year 2 contains 12 lunations. for 349;15,44°/ (29;31,50...· 0;59, 8, ...) = 12, wbereas the difference between year 3 and year 2 is 378;22,2° (the same value than for year 1), indicating that year 3 contains 131unations (378;22,2°1 (29;31.50...· 0;59, 8, ...) = 13), as is the case with year l. With this procedure, we sce that 50 is the lotal number of lunations in the ftrst 4 years, 99 in the firsl 8 years, .. _, 7,521 in the first 608 years. and so on. That ¡s, for Vímond 1 year is equivalenl to 13 mean lunations, 2 years is equivalenl lo 25 mean lunations, etc. Where possible, we have derived ¡he associated mean motioos from the entries for year 608, because those for years 1216 and 2432 are not completely legible in the manuscript. The mean motion in solar anomaly rcsulting from the entry for year 608 (Os 20;6,6°), that is, after 7,521 lunations, and the length of the synodic month obtained befare (29;31 ,50,7,44,35d ), is 0;59,S,S,23,300/d, ror

(60S· 360° + 20;6,6°)/(7521. 29;31,50,7,44,35d) ~ 0;59,S,S,23,l00Id.

This daily mean motion implies ayear length of 365; 15,42,32d which is sidereal. In the Parisian Alfonsine Tables, however, the length ofa sidereal year is variable, and the fixed length of the tropical year is 365; 14,33,9,57,...d (- 360010;59,S,19,37,19,13,56°ld). Similarly, the mean motion in lunar anomaly can be computed from the entry corresponding to 7,521 lunations (year 608 :4s S;22,2°), for 7521 lunations corresponds 1'0 S060 complete revolutions in anoma1y with an excess of about 120° (compuled with approximate values for the appropriate parameters). Hence, with lhe data in the text, the mean motion in lunar anomaly is

(S060· 360° + 12S;22,2°)/(7521 . 29;31,50,7,44,35d) = 13;3,53,57,27,11°ld, in very good agreement with the corresponding value in the Parisian Alfonsine Tables (13;3,53,57,30,21°/d); the difference only accumulates to 1° in wel! over 10,000 years. As for the motion of the solar apogee derived froro the entJy corresponding to 7,521 lunations (year 608: Os ti ;32,36°), we find O;0,0,ll,13,35°/d. By the same reasoning, the mean motioo of the lunar Ear/y Alfonsm~ Astronomy in Paris: 7ñ~ Tables 01John lIimond (1320) 221

ascending node resulting trom the entry of year 8 in the table (6s 25;27,27°) is --O;3,IO,IS.6,4Bo/d. in contrast lO the value found in the Parisian Alfonsine Tables (-0;3,1O,3B,7,14,49,IOO/d). In this case, the entry in the manuscript for 608 years is cOmJpt.

Vimond Parisian Alf. T.

Year 365; 15,42,32d 365;14,33,9,57d Solar anomaly 0;59, 8, 8,23,30o/d 0;59, 8,19,37, 19°/d Lunar anomaly 13; 3,53,57,27,11 13; 3,53,57,30,21 Solar apogee O; O, 0,11,13,35 Ascending nade -O; 3,10,18, 6,48 -o; 3,10,38, 7,14

We note that Vimond's valuc for the motion of the solar apogee includes precession as well as its proper motion for, ifwe add the value for the mean motion in solar anomaly (which is sidereal) to the motion of the solar apogee, we find 0;59,B,19.37,4°/d , in clase agreement with the corresponding value orthe mean morion in solar longitude (tropical) in the Parisian Alfonsine TabIes. In the Almagest, the planetary apogees are sidereally fixed whereas the solar apogee is tropically fixed. In the 9th century, astronomers in Baghdad fixed the solar apogee sidereally so that it too was subject lo precession (or trepidation). But in the 11th century Azarquiel realized that the solar apogee had a proper motion in addition to precession, and fixed its amount as 1° in 279 Julian years or about O;O,O,2°/d (Chabás and Goldstein 1994, p. 28). In one Andalusian tradition, this proper motion oC the solar apogee was applied to the planelary apogees as well (see Samsó and MilJás 1998, p. 269; cf. Mestres 1996, pp. 394-395). If we take al-Battanl's value for precession oC l° in 66 years or about 0;0,O,9°/d and add it to the proper motion of the solar apogee, the result is about 0;0,0,11 O/d. There is no hint oC this proper motion for either the solar apogee or the pIanetary apogees in the Parisiao Alfonsine Tables where these apogees are all sidereally fixed and, instead of precession, the Parisian Alfonsine Tables have tables for trepidation; hence, there is nothing in those tables with which to compare directiy the motion oC the solar apogee in Vimond's tables. We see, then, that the paramelers in Vimond's tables are nol identical with those in the Parisian A1fonsine Tables, and sorne oC these parameters (e.g., the length oC the solar year) are defined dilTerentiy. 222 J. Chabás and B. R. Goldstcln

ff. 1v-2r Table 4: monthly radices

This table displays the radices for the solar anomaly, the lunar 'anomaly, the solar apogee, and the lunar ascending node for 25 consecutive syzygies after the corresponding integer numbers of semi­ lunations have elapsed. An excerpt is shown in Table 4.

Table 4: monthly radices (ff. 1v-2r)

Syzygy 1 Syzygy 2 Syzygy 25 sn s (') s (')

Solar anomaly o 14;33, 9 o 29; 6,19 O 3;48,53 Lunar anomaly 6 12;54,30 O 25;49, 1 4 22;42,27 Solar apogee O O; 0, 3 O O; O, 6 O O; 1, 9 Ascending node 11 29;13,10 11 28;26,20 11 10;29,13

The entries represent the progress made by the Sun, Ihe Moan, the solar apogee, and the lunar node in 1,2, ...,25 mean semi-Iunations of 29;31,50,7,44,35d1 2 = 14;45,55,3,52,17d. The cotries in Ihis table agree with those in Table 3, for in each case the value for 26 consecutive semi­ lunations (the sum of the entries for Syzygy 1 and Sygygy 25 in Table 4) equals the value for 13 lunations (year 1 in TabJe 3).

f. 2r Table 5: Sun

This table in 5 columns is original in presentation. Column 1 gives the argument (argumentum) at 3°-intervals in signs and degrees from Os 3° to 12s 0°; this is the mean solar anomaly. Column 2 displays the true solar anomaly (motus completlls) in signs, degrees, and minutes. Column 3 (motus gradus) displays the ¡ncrement in true anomaly per degree of the argument. Column 4 gives the solar velocity, in units of minutes and seconds of arc in a minute of a day (minutum diel), i.e., in a sixtieth of a day. Column 5 displays the time (also called argumentum), in days, with sexagesimal fractions of a day, that the Sun takes to complete the arc indicated in column l. Early Alfonsjne Aslronomy in Paris: The Tables o/John Yimond (1320) 223

Table 5: Sun (f. 2r)

(1) (2) (3) (4) (5) argum. motus c. motlls g. mili. diei argum. s (0) s (0) mm. mm. d

O 3 O 2;54 57;5 I 0;57 3; 2,38 O 6 O 5;47 57;5 I 0;57 6; 5,16

227 2 24;51 59;47 0;59 88;16,18 3 O 2 27;50 59;59 0;59 91;18,55 3 3 3 0;50 60; 3 0;59 94;21,33 3 6 3 3;50 60;16 0;59 97;24,11 3 9 3 6;51 60;19 0;59 100;26,49

5 27 5 26;53 62;24 1., 179;35,13 6 O 6 O; O 62;24 l·, 182;37,51 6 3 6 3; 7 62;22 I., 185;40,29

8 21 8 23; 9 60;16 0;59 264;48,53 8 24 8 26;10 60; 3 0;59 267;51,31 8 27 8 29; 10 59;59 0;59 270;54, 9 9 O 9 2; la 59;47 0;59 273;56,46 9 3 9 5; 9 59;43 0;59 276;59,24

11 27 1I 27; 6 57;51 0;57 362;13, 4 12 O 12 O; O 57;51 0;57 365; I5,42

To obtain an entry in column 5 multiply the corresponding entry in column I by the daily mean motian in solar anomaly; the entry for 3600 (365; 15,42d) represents the length oC the sidereal year, in good agreement with the value deduced from 99 mean synodic months in Table 3. As shown in Table 5A, the difference between the argument (col. 1) and the true anomaly (col. 2) represents the solar equation, with a maximum oC 2; 100 as in the Parisian Alfonsine Tables. To emphasize the solar equation, we have added a third column for lhe djíferences between entries in columns II and 1, labeled O-L 224 J. Chabás and B. R. Goldstein

Table 5A: the solar eguatían embedded in Table 5

11 0- J argumentum motus completus s n s (') (')

O 3 O 2;54 -O; 6 O 6 O 5;47 -O; 13

2 27 2 24;51 -2; 9 3 O 2 27;50 -2;10 3 3 3 0;50 -2;10 3 6 3 3;50 -2;10 3 9 3 6;51 -2; 9

5 27 5 26;53 -O; 7 6 O 6 O; O O; O 6 3 6 3; 7 O; 7

8 21 8 23; 9 2; 9 8 24 8 26; 10 2; 1O 8 27 8 29;10 2; 1O 9 O 9 2; 1O 2;10 9 3 9 5; 9 2; 9

11 27 11 27; 6 O; 6 12 O 12 O; O O; O

The entries for the solar eguatían are not explicit in Vimond's table; they can be graphed as a smooth curve but they do not aJlow us to decide which specific tabJe fOf the solar equatían he used. The reason is that Vimond's entríes are only given to minutes in contrast to most other tables in which lhe maximum equatían is 2; 10,00 where entries are given to seconds, and rounding those values produces Vimond's entries. Eariy Alfonsíne Astronomy in Paris: The Tobles 01John Yllnond (J 320) 225

ff. 2v-3r Table 6: Moon

This table has Lhe same fonnat as Table 5. An excerpt is displayed in Table 6.

Table 6; Moon (ff. 2v-3r)

(1) (2) (3) (4) (5)

argwnenlllm mo/us c. morus mino mino die; larilud. s (0) s (0) (0) seco mm. (0)

O 1 11 29 O; 5 5 12; 9 O; 5,13 O 2 11 28 0;10 5 12; 9 O; 10,27

229 9 1 4;54 O 13; 4 4;59,58 3 O 9 O 4;55 O 13; 5 4; O, O• 3 1 8 29 4;55 O 13; 6 4;59,58 3 2 8 28 4;56 O 13; 8 4;59,50 3 3 8 27 4;56 O 13; 9 4;59,35 3 4 8 26 4;56 O 13; 9 4;59,15 3 5 8 25 4;56 O 13; 11 4;58,51 3 6 8 24 4;56 O 13;13 4;58,21 3 7 8 23 4;56 O 13;14 4;57,45 3 8 8 22 4;55 O 13;15 4;57, 4

5 29 6 1 O; 6 6 14;25 O; 5,13 6 O 6 O O; O 6 14;25 O; O, O

• Sic, instead of5;0,0.

Column 1 gives the argument (argwnemum) at 10-intervals in signs and degrees from Os 1° to 6s 0° and its complement in 360° from 6s 00 to lIs 29". For columns 2, 3, and 4, one enters with the mean argument oC lunar anomaly, whereas Cor column 5 one enters with the argument of lunar latitude. Column 2 displays the lunar equation of center (morus comple/llS) in degrees and minutes with a maximum oC 4;56° as in the 226 J. Chabás and B. R. Goldstein

Parisian Alfonsine Tables. Column 3 (molus minulI) displays [he tine-by· ¡ioe differences in column 2 divided by 60 (far purposes of interpolation). Column 4 gives the lunar velocity, in minutes and seconds, in a minute of a day (minutum diel). The minimum corresponds lO O;30,23°/h and the maximum to O;36,3°1h: fOT a comparison with other tables fOT lunar ve1ocity, see Goldstein 1996. Column 5 displays the lunar latitude. with a maximum of 5;0,0° as in the Parisian Alfonsine Tables and (he Almagest. lt is surprising that the expression motus completus is used here fOf the lunar equarion of center. whereas in Table 5 it was used fOf the true solar anomaly; c1early, it has a range of meanings and canoot be translated by a single expression.

f. 3r Tablc 7: true syzygies

There are two subtables for computing the time from mean to true syzygy: see Tables 7(1) and 7(2). The first subtable is a double-argument table where, on analogy with the other subtable, the vertical argumenl seems to be the elongation between the Sun and the Moan (at 10-intervals from 1° to 7°) and the hori~ntal argument, the velocity in elongation (i.e., the difference between the lunar and the solar velocities) in degrees per minute of a day (only four values for the velocity in elongation are given: 11,12,13, and 14).

Table 7(1): true syzygies (f. 3r)

• 11 [diff.] 12 [diff.) 13 [diff.) 14 (0) (") (0) (0) (0)

1 1;31 7 1;24 7 1; 17 5 1; 12 2 3; 2 15 2;47 13 2;34 11 2;23 3 4;33 23 4; 1O 19 3;51 16 3;35 4 6; 4 31 5;33 25 5; 8 22 4;46 5 7;35 38 6;57 32 6;25 28 5;57 6 9; 6 45 8;20 38 7;42 33 7; 9 7 10;36 53 9;43 45 8;58 38 8;20 Eariy AI/onsine AslrOllomy in Paris: TIJe Tables 01Jolm Vimond (1320) 227

*ln the MS, gradus velocitatis appears aboye this column but it refers to the headings ofthe other columns, labeled: 11, 12, 13,[4.

An entry, e, in this subtable was derived by means of the fommla (expressed in modero notation)

e ~ 16;40' ~ / [(vm(l) - v.(I)] where 11 is the true elongation at mean conjunction (or the result after subtracting 1800 at mean opposition), and the velocity in elongation, vm(t) - v,(t), is the difference between the daily velocities of the Moon and the Sun at the time of mean syzygy. We cannot give a satisfactory explanation for the factor 16;40 (= 100/6) or for the headings ofthe columns indicating that the entries are in degrees and minutes (rather than in units of time). Between these [our columns, one finds the differences, in minutes (but labeled "seconds"), between two consecutive entries in the same row, to facilitate interpolation. The second subtable is also a double-argument table giving the time in days as a funclion of the elongation (at intervals of O; Io from O; 1o to 10, or 60 minutes) and !he velocity in elongation in degrees per minute of a day (again, only 4 values for the velocity in elongation are given: 11, 12,13, and 14). Between these four columns, one finds {he differences, in minutes ofa day, between successive entries in the same row, to facilitate interpolation. Sorne selected rows of this subtable are displayed in Table 7(2). The entries in Ihis subtable were computed by means of the formula (expressed in modero notation)

where M is the time interval between mean and true syzygy, 11 is the true elongation, and !he velocity in elongation, Ym(t) - vSPtolemy in Almagest V1.4 (Chabás and Goldstein 1997, pp. 93-96; cf. Kremec 2003, pp. 305-329). Madrid, Biblioteca Nacional, MS 4238, f. 67r, reproduces both subtables except that the last row of Table 7(2) corresponds to the argument of9 mino 228 J. Chabás and B. R. Goldstein

Table 7(2): true syzygies (f. 3,) min.· 11 [difI] 12 [diff.] 13 [diff.] 14

mm. seco mm. seco mm. seco mm.

1 5;27 27 5; O 23 4;37 20 4; 17 2 10;55 55 10; O 46 9; 14 40 8;34

9 49; 5 245 45; O 208 41 ;32 178 38;34

days mm. days mm. days mm. days

10 0;55 5 0;50 4 0;46 3 0;43

59 5;22 27 4;55 23 4;32 19 4;13 60 5;27 27 5; O 24 4;37 20 4;17

·ln the MS, gradus velocitalis appears aboye this column but it refers to the headings of lhe other columns, labeled: 11, 12, 13,14.

ff. 3v-4r Table 8: correction of the lunar position for each day between syzygies

This double-argument table displays two columns for each day, from day I to day 14. The days in the horizontal argument refer to the time from conjunction to opposition. The vertical argument is given at intervals of 12°, from Os 12° to 12s O°. The heading cal1s it elongalio lune ab auge epicicli and it represents the mean lunar anomaly at mean syzygy. Fer each day, the first ceJurrm gives the increment in lunar lengitude, here caBed motus complerus, in signs and degrees, to be added lo the mean lunar longitude al the preceding mean syzygy, whereas Ihe second column displays one sixlieth of the differences between successive entries in Ihe same row, here called motus ad minuwm diei, and given in arc·minules. Early Afjonsine Ast,-ollomy in Pa,-¡s' The Tables o/John Vimolld (/320) 229

The entries in the second column thus represent the true lunar velocity in a minute oC a day for that particular day.

Table 8: correction of the lunar position for each day between syzygies (ff. 3v-4r)

Day \ Day2 Day 14 molus c. mm n/O/liS c. mm mo/us c. mm s CO) s (') s CO) s CO)

O 12 O 10;57 11 ;53 O 22;50 11 ;55 6 5;37 14;37 O 24 O \0; 7 12; 1 O 22; 8 12; 6 6 6;41 14;26 I 6 O 9;24 "12;11 O 21 ;35 12; 19 6 7;38 14; 1O

2 O O 21; 4 12;53

2 24 O 8; 10 13; 18 6 9;27 13;56

5 18 O 13;37 14;50 O 28;27 14;52 6 5;25 11 ;49 6 OO 14;45 14;45 O 29;30 14;41 6 4;27 11 ;51 6 12 O 15;47 14;35 I 0;22 14;26 6 3;30 11 ;55

7 18 1;38 12;52

8 12 O 18; 11 13; 7

9 6 5 29;30 13;28

I I 6 O 13;46 II ;49 O 25;35 II ;41 6 2; 17 14;44 II 18 O 12;48 11 ;48 O 24;36 11 ;42 6 3;27 14;46 12 O O 11;51 11 ;49 O 23;40 11 ;47 6 4;29 14;45

As mentioned above, JOM of Spira composed canons 10 sorne of Vimond's tables. In particular, the canon in Vatican, Biblioteca ApOslolica, MS Ottob.lat. 1826, ff. 152v-153r, describes the use ofTable 8, here entitled Tabilla ven' loci lune ad dies da/os post mediam 230 J. Chabás and B. R. Goldstein

coniunccionem vel opposicionem solis el ¡une. The canon ends with an explicit reference to 10hn Vimond working in Paris:

Explicit canon tabule sequentis que esl una tabularom quas composu;t Magister Johannes Vimondi. /sle autem canDil esl undecimus COIlonu", quos composuit magisrer Johannes de Spira supra tabillas predicti magistr; Johannis Parisius.

On ff. 153v-155v we find a copy ofTable 8, but in this case the entries in the second colunm (the true lunar velocity in a minute of a day) are given to one sexagesimal place. We know of only a few similar tables for the same purpose, bUI {he entries in lhcm differ from those given by Vimond. Erfurt, MS CA 2<1 388, is a 15th-centul)' manuscript which, according to Paulle (1973), contains one of the rare copies of John of Ligneres Tabule magfle. On ff. 30r-32v, there is an expanded version ofTable 8, with lhe same structure and the same columns. [n this case, the horizontal argument runs from day 1 lo day 15 and lhe column for velocity gives entries in minutes and seconds per hour which result froro lhe entries in Vimond's Table 8 by multiplying them by 2;30 (= 60L24) for conversion froro arc to time. Another example is fumished by Levi ben Gerson (d. 1344) who compiled a double·argument table, based on his own model, for finding the lunar posilion between syzygies as a function of lhe number of days since syzygy from 1 to 14 and the mean lunar anomaly at 100 ·inlervals from 0° lo 3500 (Goldstein 1974, pp. 148-149,246-254). Yet another such table is found in an anonyrnous zij in Hebrew for year 1400: lhis doub1e-argumenl lable shares the same slructure, bUI the anomaly is given at inlervals ofthe daily increment in mean lunar anomaly from day Oto day 27 (cf. Goldslein 2003, p. 166). The zij of Judah ben Verga (ca. 1470) also includes a lable wilh the same structure (Goldstein 2001, pp. 247, 269-270). Ear'y Alfonsine ASlroflomy jn Paris: The rabIes 01Jolm Yimond (1320) 231

f. 4r Radices for the planets

In a small table, the text gives the following values for the radices ofthe planets:

Mercury lIs lO; 6,10° Venus 3s 3;46,55° Mars 3s 15; 9,42° Jupiter 6s 4; 8, 5° Saturn 95 14; 0,46"

When recomputed for the instant of the mean conjunctioo 00 March 10, 1320, these radices thatdepend on the mean longitudes or mean arguments of anomaly (henceforth, simply "anomaly") confínn the use of this date as epoeh. In the case of the superior planets the radix ean be represented by the following formula:

Rx(p1anet) = ~ - A(Sun) where 4 is the mean longitude of the planet at epoch, and A(Sun) is the apogee of the Sun at that time. According to the Parisian version of the Alfonsine Tables, the mean motions for the superior planets on that day, in Toledo at 9;10 a.m. (=: 9;58 a.m. in Pans), counting from midnight, are:

Saturn 13;57, 1° Jupiter 274; 4,20° Mars 195; 5,57°

If we subtraet the value of the solar apogee for this epoch (89;56,15°) given by Vimond (f. 2r), we obtain:

Satum 284; 0,46" Jupiter 184; 8, S" Mars lOS; 9,42" in perfeet agreement with the radices given in the text. Note that using the standard Alfonsine value for the solar apogee at that time (89;23,50°) yields no agreement, confínning the author's preference for his value, 89;56,15°. The reason for subtracting the solar apogee is tha! for Vímond the planetary apogees partake in the molioo ofthe solar apogee. 232 J. Chabás and B- R. Goldstein

For Venus and Mercury, Vimond's radices can be obtained by adding the planet's anomaly and the solar longitude and subtracting from the suro the value for lhe solar apogee al epoch. For Venus we compute according lo the Parisian Alfonsine Tables al Vimond's epoch:

Rx.(Venus) = t· v + ño(Venus) + 4(Sun) - A(Sun) = 24754;52,55d· O;36,59,27,23,59,31°/d + 45;45,55,19° + 356;10,50°• 89;56,15° = 93;46,58° where t, the time from epoch Alfonso to epoch Vimond, is 24754;S2,55d; v, the mean moliao in anomaly for Venus, is O;36,59,27,23,59,31°/d; Cio(Venus), lhe radix for Venus's mean anomaly al Alfonso's time, is 45;45,55,19°; MSun), the mean longitude orlhe Sun al Vimond's epoch, is 356;10,50°; and A(Sun), the solar apogec al Vimond's epoch, is 89;56,15°. This result, 93;46,58°, differs from the radix in Vimond's text by only 0;0,3°. For Mercury, we compute according lO the Parisian Alfonsine Tables at Vimond's epoch, as for Venus, where 4(Sun) - A(Sun) = 266; 14,35°:

Rx(Mercury) = 24754;52,55d . 3;6,24,7,42,40,52°'d + 213;48,38,56° + 266; 14,35° = 346;6,15° whereas Vimond's text has lis 16;6,10° (= 346;6,10°), in cxcellent agreement with OUT recomputation. A short text below these radices tells us that we should add two quantities, the radix for the planet and the solar apogee. For Vimond the solar apogee and each of lhe planetary apogees share the same motion; hence the difference between them is always the same. in particular, since Vcnus's apogee is always the same as lhat of the Sun, nOlhing is given for Venus. The lexl then displays values for each planet of the dislance of its apogee from the solar apogee:

Satum 5s 12° = 162° Jupiter 2s 22°= 82° M ..., ls 14° = 44° Venus Mercury 3,29" = 119" Early Alfonsíne ASlro/lOmy ín París: Tire Tables 01John Vimond (/320) 233

Thcse values agree c10sely with those of Ibn IsJ:¡aq (early 13th century) [Mestres 1996, p. 395]. They are used as shifts in subsequent tables for the planets, and ean be derived [rom the radiees used in the Parisian Alfonsine Tables by subtracting the solar apogee for the time of Alfonso froro the radix of the apogee for each planet (see, e.g., the edilio prillceps of the Alfonsine Tables printed by Ratdolt (1483), e8-dl; note that the signs used there are signs of60"):

Saturo 4,2;35,20,41"-1,20;37,0"== 161;58,20" Jupiter 2,42;48,38,41" - 1,20;37,0" == 82; 11 ,38" Mars 2,4;23,51,41"-1,20;37,0"= 43;46,51" Venus 1,20;37, O' -1,20;37,0" = O" Mcrcury 3,19;51,11,41"-1,20;37,0"= 119;14,11"

These results, when rounded to the nearest degree, are in perfcet agrecment with Vimond's data. Therefore, the conclusion is that Vimond started with the same planetary apogees as those in the Parisian Alfonsine Tables.

f. 4r-v Table 9: yearly radiees

Table 9 displays seleeted entries.

Table 9: yearly radices (f. 4r-v)

Year 1 Year 2 Year 8 s n s (') s n

Mercury 4 11; 1,24 4 21; 11 ,55 2 23;56,47 Venus 8 15; 2,47 3 12;46,53 O 3;48,53 Mars 6 1; IO, 7 O 26;51,41 3 1;58,32 Jupiter I 1;53,32 2 1;19,53 8 2;52,21 Satum O 12;50,22 O 24;41,28 3 7;46,36

This tabIe displays the radices for the five planets for 1,2,3, and 4 years; for multiples of 4 years up to 76 years; and then for 152, 304, 608, 234 1. Chabás and B. R. Goldslein

1216, and 2432 years, as in Table 3. As was the case for the radices for the Sun and the Moon, 1 year is equivalent to 13 mean lunations, 2 years is equivalent lo 25 mean lunations, ...• 8 years is equivalent to 99 mean lunations, etc. The mean daily motian in longitude resulting from ¡he entries for year 8 (compuled in the same way that.was used for finding the mean motioos in Table 3) are shown below under the heading "Vimond".lfwe add the daily motion of the apogees (0;0,0,11,13,35°/d), as we did in the case of the Sun, we obtain the entries displayed in the second column, in good agreement with the values for the mean motioos in longitude in the Parisino Alfonsine Tablcs (see Ratdolt 1483).

Vimond lncluding the motion Parisiao Alf. T. oflbe apogee

Satum o; 2, 0,24, 3,56°/d o; 2, O,35,17,31°'d o; 2, O,35,17,40o'd Jupiter O; 4,59, 4, 1,19 O; 4,59,15,14,54 O; 4,59,15,27, 7 M.es 0;31,26,27,26,34 0;31,26,38,40, 9 0;31,26,38,40, 5 Venus 1;36, 7,35,47,21 1;36, 7,47, 0,56 1;36, 7,47, 1,19 Merc. 4; 5,32,16, 5,55 4; 5,32,27,19,30 4; 5,32,27,20, °

It is mast unusual for the mean motions of Venus and Mercury lO be the sum of their mean motions in anomaly and the solar mean motion, but there can be no doubt that this is what Vimond did, as is confinned by the note on f. 4vb. In fact, we know of no other medieval astronomer writing in Latin who presented the mean motions of the inferior planets in this way. For purposes of comparison, the entries for Venus and Mercury under "Parisian Alfonsine Tables" are the sum of the mean motions in anomaly and the solar mean motion: for Venus 0;36,59,27,24,oo/d and O;59,8,19,37,19°/d, and for Mercury 3;6,24,7,42,41 o/d and 0;59,8,19,37, 19°/d. Note that in Ptolemy's models lhe solar mean molion is also the mean argument ofcenter for Venus and Mercury. farly Aljo/lsine As/ronO/llY in Paris: Tlle Tables DIJo/m Vimond (/320) 235

ff. 4v-5r Table 10: monthly radices

Table 10 displays selected entries.

Table 10: monthly radices (ff. 4v-5r)

Syzygy 1 Syzygy 25 S (0) sC)

Mercury 2 0;25,26 4 0;50,53 2 10;35,57 Venus O 23;39,20 1 17;18,41 7 21 ;23,27 Mars O 7;44,14 O 15;28,28 6 13;25,52 Jupiter O 1;13,36 O 2;27,12 1 0;29,57 • Satum O 0;29,38 O 0;59,16 O 12;20,44

* Sic, instead of ls 0;39,57°.

This table displays tile radices for the five planets for 25 consecutive syzygies. The entries in this table are based on the same motions as those embedded in the previous table. As was the case for the monthly radices in Table 4, for each planet the entries for Syzygy 1 and Syzygy 25 add up to the entry corresponding to Year 1 in the previous table (except for 1" for Mercury, Mars, and Jupiter). For Venus and Mercury the mean motions extracted from Table 9 give exact agreement, confirrning the interpretation given aboye. Thus, in the cases ofVenus and Mercury one has obtained the sum of the solar anomaly and their mean anomalies, respectively, at any syzygy (see Fig. 1). This quantity is not the argument in the table ofequations (see Tables 12 and 15, below), and it is not cJear that there is any advantage to this method as against computing the mean anomaly directly. 236 1. Chabás and B. R. Goldslein

A(Sun)

Venus Ari 0°

Fig. l. A geometric interpretation of Vimond's tables for the mean motion for Venus

Tables 11 (Mercury, f. 5r), 14 (Yenus, f. 5v), 17 (Mars, f. 6r), 20 (Jupiter, f. 6v), and 23 (Satum, f. ?r): equation ofcenter and first station

Tables 11 and 12 are to be used together to compute the true longitude of a planct from its mean longitude. In mos! zijes in the Ptolemaic tradition, there is only one 5uch table for each pIanet, bul Vimond has separated those functions tha! depend on the mean argument of center from those tha! depend on the mean anomaly and pUl them in different tables. A similar idea is already found in lhe zij of fun Isbaq, described in Mestres 1996. lbn Isbaq's parameters for the maximum equations of centcr for Mars and Mercury are those of al-BauanT, but for Saturn, Jupiter, and Venus they are not; rather, they are 5;48 0 for Saturn, 0 5;41 for Jupiter, and 1;51 o for Venus. "The tables for planetary equations (...) are divided into two groups: the first group contains the tables for the equation of centre and the interpolation function. (...) The second group (two tables for each planct) contains the tables for the equations of Early Alfonsi"~ Aslronomy in Paris· Th~ Tables o[Joh" Pimond (1320) 237

anomaly at apogcc and perigee and for the middlc posilion" (Mestres 1999, p. 234). So, the arrangement of Vimond's tables bears a similarity to an AndalusianIMaghribi tradition that is not otherwise attested in Latin. However, it is not uncornmon to find later sets of tables associated with the Parisian Alfonsine Tables where Ihe planetary cquations are split into lwo tables for each planet: see, e.g., Erfurt, M8 CA Q 362, ff. 28r- 36r, where the entries are displayed al intervals of l° and the radices are given for Paris (1320) as well as for London and Bruggc (1366). Besides offering Iwo tables for the equations of each planet, Vimond's tables give additional information arranged in a presentatian which is certainly peculiar, as explained belaw.

Table 11: equation ofcenter and first station ofMercury (f. 5r)

(1) (2) (3) (4) (5) (6) s (Ol s (0) nun min mm s (0)

O 6 O 8;51 61; O 60 60 4 24;30 O 12 O 14;57 60;30 60 59 4 24;30 O 18 O 21; O 60;20 59 58 4 24;32 O 24 O 27; 2 59;40 59 57 4 24;35 1 O 1 3; O 59;30 59 54 4 24;38 1 6 1 8;57 59; O 58 51 4 24;44 1 12 1 14;51 58;50 58 48 4 24;50 1 18 1 20;44 58;20 57 44 4 24;56 1 24 1 26;34 58;10 57 40 4 25; 7 2 O 2 2;23 58; 10 57 35 4 25;20 2 6 2 8; 12 57;50 57 29 4 25;37 2 12 2 13;59 57;40 57 24 4 25;53 2 18 2 19;45 57;30 57 19 4 26; 9 2 24 2 25;30 57;30 57 14 4 26;24 3 O 3 1;15 57;30 57 10 4 26;38 3 6 3 7; O 57;30 57 6 4 26;50 3 12 3 12;45 57;30 57 4 4 27; 2 3 18 3 18;30 57;30 57 2 4 27; 9 3 24 3 24;15 57; O 56 1 4 27; 13 4 O 3 29;59 57;10 56 O 4 27; 14 4 6 4 5;40 57;30 57 1 4 27; 12 238 1. Chabas and B. R. Goldstein

4 12 4 11;25 57;30 57 3 4 27; 7 4 18 4 17; 10 57;30 57 5 4 26;59 4 24 4 22;55 57;30 57 7 4 26;46 5 O 4 28;40 57;30 57 11 4 26;34 5 6 5 4;25 57;30 57 16 4 26; 19 5 12 5 10: 10 57;40 57 21 4 26; 4 5 18 5 15:56 58; O 57 26 4 25;48 5 24 5 21 ;44 58;10 57 31 4 25;30 6 O 5 27;33 58;10 57 37 4 25; 16 6 6 6 3;22 58;30 58 41 4 25; 3 6 12 6 9; 13 59; O 58 45 4 24;54 6 18 6 15; 7 59; 10 58 49 4 24;48 6 24 6 21; 2 59;30 59 52 4 24;42 7 O 6 26;59 59;50 59 55 4 24;37 7 6 7 2;58 60;30 60 57 4 24;34 7 12 7 9; 1 60;40 60 59 4 24;31 7 18 7 15; 5 61; O 60 60 4 24;30 724 7 21;11 61;40 61 60 4 24;29 8 O 7 27;21 61 ;50 61 60 4 24;29 8 6 8 3;32 62; O 61 59 4 24;29 8 12 8 9;44 62; 1O 61 59 4 24;30 8 18 8 15;57 62;20 61 58 4 24;32 824 8 21; 11 62;30 62 57 4 24;34 9 O 8 28;26 62;40 62 56 4 24;36 9 6 9 4;44 63; O 62 55 4 24;38 9 12 9 11; 2 63;10 62 54 4 24;39 9 18 9 17;21 63;10 62 54 4 24;40 924 9 23;44 63;20 62 53 4 24;41 10 O 10 O; 3 63; 10 62 53 4 24;42 10 6 10 6;22 63; 10 62 53 4 24;41 10 12 10 12;41 63; 1O 62 54 4 24;40 10 18 10 19; O 63; O 62 54 4 24;39 10 24 10 25; 18 62;50 62 55 4 24;37 11 O 11 1;35 62;40 62 56 4 24;35 11 6 11 7;51 62;40 62 57 4 24;33 11 12 11 14; 7 62;10 61 58 4 24;31 11 18 11 20;20 62; O 61 59 4 24;30 11 24 11 26;32 61;50 61 60 4 24;29 12 O 12 2;43 61;20 60 60 4 24;29 &rly A/fonsjne ASlronomy in Poris: The Tob/es 01John Villfond (l310) 239

The table for the equation of center of each of the five planets has six columns. Column I gives the argument (argumentll1l1) at 6°·intervals in signs and degrees froro Os 6° to 12s O°. Column 2 displays the entry in coL 1 corrected for the equation oC center (moflls completlls), in signs, degrees. and minutes. The author follows here the same paltem as that for the true solar anomaly (see Table 5). Column 3 (motlls gradlls) gives the increment of the true argument per degree of the argument. in minutes and seconds. Most entries in this column are generatcd by dividing by 6 the differences between two successive entries in col. 2 and thus were probably ¡ntended for interpolation in col. 2. Column 4 (motus diei) displays the velocity in minutes of arc per day, and the rangc of values for each planet is the same as in the column labeled mOfllS centri or motus pllncti (that only depends on the argument of center) in the table for planetary velocities associated with the Toledan Tables and the Castilian Alfonsine Tables (Chabás and Goldstein 2oo3a, pp. 170-182); for the other component oC the planetary velocity, see Tables 12, 15, 18, 21, and 23, col. 4, below. So, the entries in this column are only one component ofthe planet's velocity. Column 5 is intended to provide minutes of interpolation and is headed diametri (perhaps lo distinguish these "linear" minutes from minutes of an hour. minutes ofa day, and minutes ofa degrec). Finally, column 61ists the first station in signs, degrees. and minutes.

Table 14; equation ofcenter and first stalion ofVenus (e. 5v)

(1) (2) (3) (4) (5) (6) s (0) s (0) nun mm mm s (0)

O 6 O 5;47 57;50 57 O 5 15;52 O 12 O 11;34 57;50 57 O 5 15;54

2 24 2 21;51 59;50 59 27 5 17; 2 3 O 2 27;50 60; O 59 31 5 17; 11 3 6 3 4;50 60;20 59 33 5 17; 17 3 12 3 9;52 60;30 60 36 5 17;23 240 J. Chabás and 8. R. Goldslcin

8 18 8 20; 8 60;20 59 36 5 17;23 8 24 8 26; 10 60; O 59 33 5 17; 17 9 O 9 2; 1O 59;50 59 31 5 17; 11 9 6 9 8; 9 59;30 59 27 5 17; 2

11 24 11 24; 13 57;50 57 O 5 15;52 12 O 12 O; O 57;50 57 O 5 15;50

Table 17: equation ofcenter and first statiol1 of Mars (f. 6r)

(1) (2) (3) (4) (5) (6) 'COl , el mm mm mm , (0)

O 6 O 12;31 50;50 26 6 5 8;41 O 12 O 17;36 50; O 26 4 5 8;21

12 12;22 49; O 26 O 5 7;29 18 17; 16 49; 1O 25 O 5 7;31

4 18 4 6;36 60;30 31 32 5 13;46

7 12 7 11 ;33 73;30 38 60 5 19; 14 7 18 7 18;54 73; 1O 38 59 5 19; 13

10 12 10 23;24 58;50 30 31 5 13;36

11 24 12 2;13 51;50 27 10 5 9;31 12 O 12 7;24 51; 10 26 8 5 9; 6 Ear/y Alfollsine Aslrol1omy i/J Paris: nle Tables o/Jolm Vimolld (/320) 241

Table 20: equation ofcenter and first station of Jupiter (e. 6v)

(1) (2) (3) (4) (5) (6) s e) s (0) min mm min s (0)

O 6 O 11 ;43 58; O 5 23 4 5; 19 O 12 O 17;31 57;20 5 20 4 5; 9

2 12 2 12;59 44; 10 4 O 4 4·, 6 2 18 2 18;23 44; O 4 O 4 4; 5 2 24 2 23;48 44; O 4 O 4 4; 5 3 O 2 29; 12 44; 10 4 O 4 4; 6

5 24 5 18; 3 60; 10 4 32 4 5;44

8 12 8 10;55 66;20 6 59 4 7; 1O 8 18 8 17;33 66;30 6 60 4 7; 11 8 24 8 14; 14 66;20 6 60 4 7; 11 9 O 9 0;52 66;20 6 60 4 7; 1O

11 18 11 23;57 59;50 5 32 4 5;47 11 24 11 29;56 59; 10 5 29 4 5;39 12 O 12 0;51 * 58;40 5 26 4 5;29

* Sic, instead of 5;51.

Bu! for a shift of the entries, the equations of center for Mercllry, Mars, and Saturo that can be derived from cols. 1 and 2 are basically tbe same (with minor variants) as in the zij of al-SattanT (Nallino 1903-1907, 2:110-137) and the Toledan Tables (Toomer 1968, pp. 60-68; F. S. Pedersen 2002, pp. 1259-1308). 242 J. Chabás and B. R. GoldSJein

Table 23: equation ofcenter and first statian ofSaturo (f. 7r)

(1) (2) (3) (4) (5) (6) s (0) s (0) mm mm mm s (0)

O 6 O 8;46 66;20 2 57 3 25;22 O 12 O 15;24 66; O 2 55 3 25;19

2 12 2 18;31 59;20 2 31 3 24; 11

5 6 5 6;39 53;30 2 O 3 22;45 5 12 5 12; O 53;30 2 O 3 22;44 5 18 5 17;21 53;40 2 O 3 22;45

8 12 8 5;29 60; 1O 2 31 3 24; 11

11 6 11 12; O 67; 1O 2 60 3 25;30 11 12 11 18;43 67; 1O 2 60 3 25;28 11 24 11 25;25 67; O 2 59 3 25;27 12 O 12 2; 7 66;30 2 58 3 25;25

The maximum value for Mercury (3;2") occurs at about Os 24° and 756°, that ofMars (11 ;24°) al 45 18° and lOs 12", and that ofSaturo (6;31") a12s 12° and 8s 12". However, for the other two planets the entries differ systematically fram those in the above~mentioned zijes: for Venus the maximum value is 2; 10° al 35 O" and 35 6°, and 8s 24° and 95 O"; and for Jupiter the max.imum value is 5;57° al 55 24" and lIs 18°. The entríes for Mercury, Mars, Jupiter, and Saturo are shifted by about 119°, 44°, 82°, 0 and 162 > respectively, in rclarían lo those in the zij of al-BattanT and the Toledan Tables. No such shift appears in the table for Venus. As mentioned above, these shifts resull from the difference between the apogee of each of the planets and that of the Sun. Because of these shifts, for the superior planets one enters these tables in col. 1 directly with their mean motions for a given syzygy (the radix plus the motion in years and semi-Iunatjons); for Venus and Mercury one enters with the solar anomaly for a given syzygy. Clearly, Vimond intended to make this table more &r/yAlfonsine Astrollomy in Paris: The Tabla allonn Jlimond (/J20) 243

"user-friendly" than the standard version oC the table Cor lhe equation oC center. Vimond has a double motion of the solar apogee: precession and proper motion. The planeta!)' apogees are fixed with respect to the solar apogee (Le., they are subject to both precession and lhe proper moHon of lhe solar apogee). lfwe add the solar apogee (about 90°) to the values for lhe shifts listed aboye, we find that the planeta!)' apogees are 209° for Mercury, 900 for Venus, 135° for Mars, 172° for Jupiler, and 252 0 for Saturno ln the Toledan Tables, {he apogees of the Sun and of Venus are both 77;50° (Toomer 1968, p. 45), that is, about 12° less than 90°. Adding this diffcrence to the planetary apogees in the Toledan TabIes rounded lo degrees, we find the following:

Apogees

V (from the shifts) V (froro the radices) 11 + 120

Mercury 209° 2090 210° Venus 90' 90' 90' Mars 135° 134° 134° Jupiter 172° 172° 1760 Satum 252 0 252° 2520

The agreement of Vimond's data with the apogees in the Toledan Tables shows that Vimond has included the mOlion of the solar apogee in the motions of the planeta!)' apogees, thus following a theo!)' for which there was no previous evidence outside al-Andalus and the Maghrib (SanlSÓ and MiIlás 1998, pp. 268-270). We know oC no other set of planetary equation tables arranged in this way. See also Table 27 (equation ofaccess and recess), below, for yet another shift in Vimond's tables. The maximum values for the equation of center in Vimond's planetary tables are the same as in the editia prillceps of the Alfonsine TabIes (see Table IIA). Despite their agreement for the values of the maximum equations, the structure oC Vimond's tables is ve!)' different from that oC the Parisian AIConsine Tables and would seern to be independent oC it. Moreover, it is significant that the maximum equation oC center for Jupiter in both cases is 5;57°, for this value is not known in any text prior to the Parisian Alfonsine Tables, indicating a strong connection between the tables oC 244 J. Chabás and B. R. Goldslein

Vimond and the work of his Parisian contcmporaries. The oogm or derivation of this parameter for Jupiter is nol described in any extanl text, and it is Iikely that this value was simply taken from an earlier work: the mast reasonable candidate is the Alfonsine Tables as they existed in Castile.

Table IlA: maximum values for the equation ofcentcr

al-Baltan! Tolerlan T. Vimond Parisian Alf. T.

Mercury 3; 2° 3; 2° 3; 2° 3; 2° Venus 1;590 1;590 2; 10° 2; 1O" Mars 11;24° 11;24° 11 ;24 0 11 ;240 Jupiter 5; I5° 5; t5° 5;57° 5;57° Saturn 6;31 0 6;31 0 6;31 0 6;31 0

For all plancts, excepl Mercury, an entry, e, in column 5 can be computed, bUI for shifts. [rom the modem formula

e = 60 (1 - cos ;()12, where Kis the mean argument of center. The same approach is [aund in Levi's lunar theory (Goldstein 1974, table 35, col. 11: see p. 54). The entries for Mercury in col. 5 do not follow the same pattem as that for the rest of the planets. The entries can be recomputed, approximately, according to the following fonnula:

C,(K) ~ [D - r(K)JI [D - dJ (1] where D is the maximum dlslance of the center of the epicycle from the observer, d is the minimum distance, and r(K) is the distance as a function of the mean argument ofcenter, K. Eady Alfonsine AslrOllomy ill Paris: The Tables o[Jolm Vimol/d (1320) 245

.... ~e " o .--o. _00 E" ...,::l ~ ~

V> ::8 ::l c: ~ -~

'D o o o o M '" minutes ofproportion

Fig. 2. Vimond's equation of center, col. 5, for Venus, Mars, Jupiter, and Satum 246 J. Chabás and B. R. Goldstein

g N

o '"

o o minutes ofproportion

Fig. 3. Vimond's equation ofcenter, col. 5, for Mercury EarlyAlfonsine Aslronomy in Paris: The TablesofJohn Vimond (1320) 247

A similar formula for interpolatíon was already used by Mabash in the 9th century (as-Saleh 1970, pp. 137-138). In Ptolemy's model for Mercury D is 69 for argument 0°, d is 55;34 for an argument close to 120°, and r(l800) Is 57 (O. Pedersen 1974, pp. 313-324). Hence, formula [1] can be replaced by

C,(K) = [69 - '(K)] 113;26. (2]

In general, the computation ofr(K) is a difficult and lengthy procedure, and il is likely that Vimond (or his source) used approximations (if this, indeed, \Vas the fom1Ula he had in mind). We compuled the disrances from the observer ro the center of Mercury's epicycle according lO formulas in modem tenns given by O. Pedersen (1974, p. 320, equations 10.34 and 10.35), and lhen used them in equation [2], aboye. A comparison of our results for CS(K) with the entries in Vimond's table is displayed in Table liB. Col. Ir has Ihe values for cs(i() that depend on the distances computed according lo the formulas given by O. Pedcrsen and equ. [2], aboye; col. m has the argumellls in Vimond's table (with the shift); and col. IV has rhe entries in Vimond's Table 11, col. 5. Although the agreerncnt is not exacl belween col. U and col. IV, the trend is cleaT. Vimond's value for 180°,37, has the poorest agrcemcnt, bul Ihis entry should probably be corrected to 36, judging from the surrounding values.

Table llB: a comparison ofcalumn 5 for Mercury wirh recomputation

11 IIJ rv

K Cs(K) K(Vimond) cs(i<): Vimond O o; 120 O 30 10;50° 150 11 54 29;24 174 31 60 34; 15 180 37 66 38;53 186 41 90 53;33 210 55 120 60; O 240 60 150 56;38 270 56 ISO 53;34 300 53 248 J. Chabás and B. R. Goldstein

It may be of¡ntereSI IhaI in Copemicus's table for lhe equations of Mercury (Copemicus 1543, fr. l77v-l78r), his col. 4 (rar interpolatíon) shows lhe same trend as Vimond's col. 5. We are convinced lhal column 5 in Vimond's tables for Ihe equation of centcr is ¡ntended to be used for interpolalion with column S in the tab1es for the equation of anomaly, and Ihis is analogous to Copemicus's lIse of his col. 4 (sce below). lndeed, Vimond's col. 5 serves much lhe same purpose as col. 8 in Ptolerny's tables for lhe planelary equations (Almagesl, XI.Il) bUI, since lhe definitions for the columns thal yield lhe cquation of anomaly are diffcrent, so is lhe funetían for interpolation. Moreover, in contrastto the geometric methods in lhe Almagesl used for computing the coefficients of interpolation for each of the four planets (Venus, Mars, Jupiter and Satum), Vimond has approximaled the results lhal would be derived from the geometry of the models by introdueing a smgle trigonometrie funetion in those cases. In Almagest, Xl.JI (Toomer 1984, pp. 549-553), col. 8 in [he planetary equation tables is intended for inlerpotalion as a funetion of K, the mean argument of center, and the entries are given to minutes and seconds (for Ptotemy's method oC computation and a graph oC the entries in his col. 8, see Neugeballer 1975, pp. 184-186, 1267). A similar set oC vallles, given only lO minutes, is found in al-BattanT's zij in the tables for lhe planetary equations, col. IV (NaBino 1903-1907, 2: 110-137), and in corresponding tables in the Parisian Alfonsine Tables. col. 3 (Ratdolt 1483, e7r-g5v). As for the entries for the first stalion of each planet, rhey are essenlially Ihe same as In previous tables oC lhe same kind (Almagest, Handy Tables, al-Khwarizmi, al-Bat1.ii.ni, and the Toledan Tablcs) wilh lhe same shifis that we noled aboye.

Tables 12 (Mercury, r. 5r), 15 (Venus, r. 5v), 18 (Mars, r. 6r-v), 21 (Jupiter, f. 7r), and 24 (Saturn, f. 7v): equalion ofanomaly

The tables for rhe equation oC anomaly for each of the five planets have seven columns. Table J2 displays a selection of values for Ihe equation ofanomaly for Mercury. Early Alfollsille AsrrOllomy in París: The rabIes 01John Vimond (1320) 249

Table 12: cquatiol1 ofanomaly for Mercury (f. 5r)

(1) (2) (3) (4) (5) (6) (7) s (") s (") (") mm mm min see see

O 6 11 24 1;28 15 45 O; 18 2 8 012 11 18 2;56 15 45 0;33 2 8 O 18 11 12 4;24 14 44 0;48 2 8 024 11 6 5;50 14 44 . 1; 3 2 8 1 O 11 O 7; 15 14 42 1; 18 2 8 1 6 1024 8;37 13 42 1;33 2 8 1 12 10 18 9;58 13 40 1;48 2 8 1 18 10 12 11; 15 12 39 2; O 2 8 1 24 10 6 12;30 11 36 2; 18 3 9 2 O 10 O 13;39 10 34 2;35 3 9 2 6 924 14;44 10 31 2;53 3 9 2 12 9 18 15;44 9 28 3; 1O 4 8 2 18 9 12 16;38 8 25 4;14 4 8 224 9 6 17;25 7 19 4;30 4 8 3 O 9 O 18; 4 5 15 4;45 4 8 3 6 924 18;34 3 9 4;57 4 8 3 12 8 18 18;53 1 3 5; 5 4 8 3 18 8 12 19; 1 1 3 5; 1O 3 6 3 24 8 6 18;56 3 9 5;13 2 4 4 O 8 O 18;39 6 18 5; 6 1 3 4 6 724 18; 4 9 27 4;55 1 1 4 12 7 18 17; 12 11 35 4;29 O 1 4 18 712 16; 4 15 45 4;55 2 6 4 24 7 6 14;36 18 55 4;12 4 13 5 O 7 O 12;49 21 66 4;29 6 18 5 6 624 10;42 24 74 3;55 7 22 5 12 6 18 8; 18 26 81 3;12 9 29 5 18 612 5;42 28 87 2; 15 11 34 5 24 6 6 2;53 29 89 1; 10 12 36 6 O 6 O O; O 29 89 O; O 12 36 250 J. Chabiis and B. R. Goldstein

Column 1 gives thc mean argument of anomaly (argumemum) al 6°-intervals (at 3°-intervals for Mars and Venus) from Os 6° lo 65 0° and its complement in 3600 from 65 0° lo lis 24°. Column 2 displays the correclion due lo the argument of anomaly al maximum distance (mo/us completlls) in degrees and minutes and represents the difference betwcen the equation of anomaly and the correction for maximum distance (cf. Almages/, Xl.lI, columns 6 and 5; and Neugebauer 1975, pp. 183-184). The only other text of which we are awarc that treats the equation of anomaly in this way is the zij of lbn al-Banna' (d. 1321) where this presentatian is applied in his tables for Saturo and Jupiter bUl 001 in those for the orher planets (see Samsó and Millás 1998, pp. 278-285). The e;ll;tremal values in col. 2 that appear in the text are shown below; they are followed by the corresponding entries for col. VI and col. V in the zij of al-Battani(Nallino 1903-1907,2:109-137):

Vimond al-Battani

Mercury 19; 1° (= 2I ;59° - 2;58°) al3s 18° Venus 44;49° (= 45;59° - 1; I 0°) al4s 15° Mars 36;44° (= 40;58° - 4; 14°) at 4s 6° Jupiter 10;34° (= 11; 3° - 0;29°) al 3s12° Saturo 5;53° r= 6;12°-0;19°)aI3s00and (= 6; 13° - 0;20°) al 3s 6°

These corrections agree with those that follow from the Almagest as weil as Ihe zij ofal 4 Battani, the Toledan Tables, and the editio princeps ofthe Alfonsine Tables (with minor variants: 40;59° rather than 40;58° for Mars; 6; 12° and O; 19° correspond to 3s 0° rather than 3s 1° for Satum), and this means that Plolemy's eccentricities undcrlie them even though, in the case of Venus and Jupiter, the eccentricities \Vere modified for computing the equation ofccnler (cf. North 1976,3:196). Similarly, in the tables of rbn al-Baona' the eccentricities underlying the equalions of anomaly are taken from the Almagest, bUI his maximum equations of center for Venus and Jupiter are not those ofeither Ptolemy or of Vimond (Samsó and MilIás 1998, p. 276). Column 3 (morus gradlls) gives the increment of the morus comp/eJlIs in col. 2 per degree of the argument in minutes: in mosl cases the entry results from taking the difference between successive entries in col. 2 and dividing that difference by 6 (or by 3 for Mars and Venus); the Ear/y A/fol1!líne ASlrO/lomy ín París: TlJe Tabies o/JoJm Vimond (/320) 251

purpose of Ihis column is facilitate interpolation. Column 4 (motus diei) displays Ihe velocity in minutes of arc per day, and Ihe range of values for each planel is the same as in Ihe column labeled motus argumellti (that only depends on Ihe argument of anomaly) in the table for planetary velocities; see the comments to Ihe Castilian Alfonsine Tables, chapter 27 (Chabás and Goldstein 2003a, p. 170-182). So, an entry in this column is Ihe second component of the planet's velocity and it complemenls Ihe first component already displayed in Tables 11, 14, 17,20, and 23, aboye. The entries in column 5 (millu/um diametri), in minutes and seconds, actually represent degrees and minutes, and result from adding Ihe correction for maximum distance lo Ihe correction for minimum distance (columns Cs and C7 in Almagest, Xl. 11). For the extremal values in col. 5 in the texl see beJow; they are followed by tbe corresponding entries for col. V and col. VD in the zij ofal-BaltanT (Nallino 1903-1907,2: 109-137):

Vimond al-BattanT

Mercury 5;13 0 4;560 * 0 0 0 0 Venus 3;34 (= 1;42 + 1;52 ) al 5s 12 Mars 13;37 0 (~ 5;34° + 8;3°) al 5s 9° 0 0 0 Jupiter 1; 3° (= 0;30 + 0;33 ) at 3s 24 Satum 0;460 (= 0;21 ° + 0;26°) al 3s 12°

* In al-BatHinT's zij for 3s 24 0 we find 3;40 + 1;52 0 =4;56°, whereas for 3s 240 in Vimond's table we find 3; 12 0 + 2; 1° 0 = 5;13 , al-BattanT's maximum which occurs al 4s 1O°-4s 0 12 .

In the absence of instructions by Vimond it is not easy to decide ho\V ¡he correction lo the planet's mean longitude is to be computed. but il seems likely thal one componenl of this correction is lO be computed by adding an enlry in col. 2 to an interpolation factor times an entry in column 5, as is ¡he case with the tables of Ibn al-Banna' for Saturo and Jupiter. The most like1y candidate for this interpoJation factor is col. 5 in Tablc 11, for it depends on the argument ofcenter as il should (see Samsó and Millás 1998). Column 6 (motus gradus) seems to be the increment per degree ofargument orthe entries in col. 5: in many cases Ihe entry in col. 6 results from taking the difference between successive entries in col. 5 and dividing it by 6 (or by 3 for Mars and Venus), and it is for purposes of inlerpolation. The entries in col. 6 are given in seconds. The entries in 252 J. Chab:is and B. R. Golcistein

colutTUl 7 (mOlus diel) are also given in seconds; lhey are probably associated with those in the preceding column, for in al1 cases columns 6 and 7 have their extremal values for the samc arguments, but we have failed lO identify their specific purpose.

Table 15: equation ofanomaly for Venus (f. 5v)

(The entries from Ss 18° to 65 121) are given al 2°-intervals, rather than al 3°~intervaJs as in the rest ofthe table.)

(1) (2) (3) (4) (5) (6) (7) s (0) s (0) (0) mm mio mio sec sec

O 3 11 27 1; 15 25 15 O; 2 OO O 6 11 24 2;30 25 15 O; 3 1 O

4 12 7 18 44;44 2 1 2; 18 2 1 4 15 7 15 44;49 2 1 2;25 2 1 4 18 712 44;44 6 4 2;32 3 2

5 12 6 18 33;25 74 46 3;34 2 1 5 15 6 15 29;43 89 55 3;27 4 2 5 18 612 25;25 104 64 3; 14 7 4

5 28 6 2 4;48 144 89 0;45 22 14 6 O 6 O O; O 144 89 O; O 22 14 Early Alfonsine AslronOIllY iJl Paris: The Tables ofJohn Villlond (/320) 253

Table 18: equation ofanomaly for Mars (f. 6r-v)

(The entries from 5s 180 to 6s 12° are given al 2°~intervals, rather than at 3°-intcrvals as in thc rest of the table.)

(1) (2) (3) (4) (5) (6) (7) s (0) s (0) (0) mm mm mm sec sec

O 3 11 27 1; 8 23 11 O; 8 3 O 6 11 24 2; 16 23 11 O; 17 3

4 3 727 36;40 1 1 8;53 9 4 4 6 724 36;44 OO 9; 19 9 4 4 9 721 36;43 3 1 9;46 9 4

5 6 624 28; 15 46 21 13;30 OO 5 9 6 21 25;56 53 25 13;37 6 2 5 12 6 18 23; 17 62 29 13; 19 13 6

5 28 6 2 3; 1 90 42 2;29 74 35 6 O 6 O O; O 90 42 O; O 74 35

Table 21: equation of anomaly for Jupiter (f. 7r)

(1) (2) (3) (4) (5) (6) (7) s CO) s (0) (0) mm mm mm sec sec

O 6 11 24 0;57 9 8 O; 4 012 11 18 1;52 9 8 O; 8

3 6 824 10;33 OO 0;59 OO 3 12 8 18 10;34 1 1 1; I OO 3 18 8 12 10;29 2 2 1; 2 O O 3 24 8 6 10; 15 3 3 1; 3 O O 4 O 8 O 9;54 5 4 1; 2 O O 254 J. Chabás and B. R. Goldstein

5 24 6 6 1;21 13 12 o; 9 6 O 6 O O; O 13 12 O; O

Table 24: equatían ofanomaly for Saturo (f. 7v)

(1 ) (2) (3) (4) (5) (6) (7) s (") s (") (0) mm mm mm sec sec

O 6 1124 0;34 5 5 O; 3 O 12 11 18 1; 7 5 5 O; 7

2 24 9 6 5;46 1 1 0;41 OO 3 O 9 O 5;53 OO 0;42 OO 3 6 824 5;53 OO 0;44 OO 3 12 8 18 5;51 1 1 0;46 O O

5 24 6 6 0;42 7 7 O; 7 6 O 6 O O; O 7 7 O; O

Figure 4 displays Plolemy's modcl for the Ihree superior planets and Venus. O is the observer, O is the center orlhe deferent circle RAe, and E is the equant point, such that the eccentricity, e = OD = DE. A is the planet's apogee, and i( = angle AEe, ¡he mean argument of center, is measured from it lO the center of the epicycle about point E. Angle OCP is the mean argument ofanomaly, a, and the plnne! is al point P. Angle HCG is lhe equatían ofcenter and il is also applied lO corree! the mean argument of anomaly lO yield the true argument of anomaly, a = angle Hep. In the case of the superior planets, ep, the direetioo_ from ¡he center of the epicycle to the planet, is always parallel to O S, the direction from the observer to the mean Sun. In the case oC Venus, EC is parallel to the direction from the observer to the mean Sun. The goal is to find the direction from the observer to the planet, Le., angle ROP is the longitude oC the planet, and R is in the direction lO Aries Oo. Eady Alfonsine Astronomy in Paris: The Tables 01Joh" Jfimond (1320] 255

A H G a K R E Ari 0° P D O

s

Fig. 4. Ptolerny's model for the lhree superior planets and Venus (not to scale)

The mean argument of center, K, is angle AEC, and the true argurnent ofanomaly, a, is angle I-ICP. With these arguments, Kand a, we can dctennine thc cquation of anomaly, c(a.), wilh Vimond's tables and compare the result with computations based on the Parisian Alfonsine Tables. According to our understanding of Vimond's procedure,

e(ex) ~ ",(ex) +e,(')· e,(ex), where Cl refers to the i-th column in the tablc. Note that cs(iC) is taken from the table for the equation of center (with the shifts), and cs(a) is taken 256 1. Chabás and B. K GoldSlcin from the table for the equation of anomaly. For instance, for Venus, when K= 120 0 and a = 135°

e(a) = e,(135°) + e,(\200). e,(\]50) e(a) = 44'49°, + 0-45, . 2'25°, e(a) = 46;38°.

With the same arguments for Venus in the Parisiao Alfonsine Tables, we find

e(a) = e,(a) + e,(K)' e,(a) e(a) = e,( 135°) + e,(1200) . e,(\ 35°) e(a)=45"59°+0"31·,,,¡-15° e(a) = 46;38° and Ihis is exactly what resulted [rom Vimond's tables. In the tables for the planetary equations in Almagest Xl.l1 and its dcrivatives in al-Battanl and in the Parisiao Alfonsine Tables (among others). the rules for computing the equation of anomaly require careru} attention lo algebraic signs. Vimond simplified Ihe rules for Ihis computatían, making his tables more "user-friendly". A similar procedllre is described by Copemicus for using his planctary tables in De revolulionihus, V.23, to compute the equation of anomaly (Copemicus 1543, ff. 173v-179r; cf. Swerdlow and Neugebauer 1984, p. 453).

Tables 13 (Mercury, f. 5r), 16 (Venus, f. 6r), 19 (Mars, f. 6v), 22 (Jupiter, f. 7r), and 25 (Satum, f. 7v): planetary latitudes

The tables for the planetary latitudes, both for the superior and lhe inferior planets, are in the style of Almagesl Xill.5, the zij of al-Battan! (NaBino 1903-1907, 2:140-141), and sorne tables associated with the Toledan Tables (Toomer 1968, pp. 71-72; F. S. Pedersen 2002, pp. 1322­ 1326), as opposed to those in the Halldy Tables and those in the zij of al­ KhwarizmT. The table for the planetary latitudes of Mercury has seven columns; the table for Venus lacles thc seventh; and the tables for the superior planets have only five columns (Le., cols. 1, 3, 4, 5, and 6). &Ir/y Alfo"si"e Astro"omy i" Par/S: The rabies o/Jole" Yimomi (1120) 257

Table 13: latitude of Mercury (f. 5r)

(1) (2) (3) (4) (5) (6) (7) , (0) mm mm min· min min· ,ce

O 12 13 57 1;44 17 O; 12 1 O 24 4 60 1;40 5 0;44 4 1 6 5 60 1;39 7 1; 6 7 1 18 14 57 1; 16 19 1;26 9 2 O 23 51 0;59 31 1;44 10 2 12 31 44 0;38 41 2; O 12 224 37 34 O; 16 49 2; 14 13 3 6 41 23 0;15 55 2;25 14 3 18 44 11 0;48 59 2;29 15 4 O 45 O 1;25 60 2;29 15 4 12 43 13 2; 6 58 2;20 14 424 40 25 2;47 54 2; O 13 5 6 36 36 3;26 48 1;29 9 5 18 29 45 3;54 39 0;48 5 6 O 22 52 4; 5 29 O; O O 6 12 13 57 3;54 17 0;48 5 624 4 60 3;26 5 1;29 9 7 6 5 60 2;47 7 2; O 12 7 18 14 52 2; 6 19 2;20 14 8 O 23 51 1;25 31 2;29 15 8 12 31 44 0;48 41 2;29 15 8 24 37 34 O; 15 49 2;25 14 9 6 41 23 O; 16 55 2;10 13 9 18 44 11 0;38 59 2; O 12 10 O 45 1 0;59 60 1;44 10 10 12 43 13 1; 16 58 1;26 9 10 24 40 25 1;30 54 1; 6 7 11 6 36 36 1;40 48 0;44 4 11 18 29 45 1;44 39 O; 12 1 12 O 22 52 1;46 29 O; O O

• Oespite the headings, these columns display degrees and minutes. 258 1. Chabás and B. R. Goldstein

Table 16: latitude ofVenus (f. 6r)

(1) (2) (3) (4) (5) (6) s (0) mm mm min· mm min·

O 12 10 12 1; 1 59 O; 16 024 9 24 0;59 55 0;33 1 6 8 35 0;55 48 0;49 1 18 7 44 0;46 40 1; 5 2 O 5 52 0;35 30 1;20 2 12 3 57 0;29 18 1;35 2 24 1 60 0;18 6 1;50 3 6 1 60 0;10 6 2; 3 3 18 3 52 0;32 18 2; 15 4 O 5 44 0;59 30 2;25 4 12 7 35 1;38 40 2;30 424 8 . 24 2;23 48 2;28 5 6 9 12 3;44 55 2; 12 5 18 10 O 5;13 59 1;27 6 O 10 12 7; 12 60 O; O 6 12 10 24 5;13 59 1; 12 624 9 35 3;44 55 2;28 7 6 8 44 2;23 48 2;30 7 18 7 52 1;38 40 2;25 8 O 5 57 0;59 30 2; 15 8 12 3 60 0;32 18 2; 3 824 1 60 0;10 6 1;50 9 6 1 57 0;19 6 1;35 9 18 3 52 0;29 18 1;20 10 O 5 44 0;35 30 1; 5 10 12 7 44 0;46 40 O; 5 •• 10 24 8 35 0;55 48 0;49 11 6 9 24 0;59 55 0;33 11 18 10 12 1; 1 59 O; 16 12 O 10 O 1; 3 60 O; O

• Despite the headings, these columns display degrees and minutes. •• Sic. Early Alfonsine Aslronomy in Paris: The Tables 01Jofm Vimond (/320) 259

In all cases column 1 displays the argument (argumenlum) at 12°_ intervals from Os 12° to 12s O°. Column 2 is only found in the tables for the inferior planets and the entries are given in minutes. The heading is radix meridiollalis in the case of Mercury and radix septenlrionalis in that for Venus. This colunm is for detennining the deviation, otherwise caBed the third component of latitude, that is, the inclinalion of the plane of the deferenl with respecl lo Ihe ecliplic. The entries for deviation can be derived from:

~3 = -0;45 . Cs for Mercury ~3=+O;10' Cs forVenus where Cs is the column for the minutes of proportion in the lable for planetary latitude in Almagest Xill.5 (given there in minutes and seconds). As will be seen, column 5 for Venus in Table 16, given only to minutes, corresponds to Cs in Almagest XIll.5. It is noteworthy that column 2 for Mercury is shifted downwards aboul 119° whereas there is no shift in the case ofVenus. This is exactly the same (eature we noticed in the tables for the equation ofcenter and the amount of Ihe shift is the same. The colurrm for deviation is certainly not a common feature in medieval tables (for a survey of the few that have them, see Goldstein and Chabás 2004), and Vimond's is Ihe earliest set of lables in the West we know to display such columns. For the inferior planets, columns 3 and 5 (diametri) give the minutes of proportion for the inclination and the slant, respectively. We note that columns 3 and 5 for Mercury also exhibit a shift of less than 120°, and thal no shifts appear in the case of Venus. We also nole that column 5 for Venus lists the rounded values in the column for lhe sixtieths found in the corresponding table in the Almagesl Xill.5, the zij of al­ Battan!, etc. For the superior planets, columns 3 and 5 give the minutes of proportion for the northem and southern latitudes, respectively, of the planets. Only half of lhe columns are filled with numbers, the others have capitalletters indicating "North" {S] and "South" [M]. Column 3 is shifted about 45° (Mars), about 100° (Jupiter), and about 110° (Satum) in relation to the corresponding columns in the Almagest, whereas the shifts for colunm 5 are increased by 1800 in each case. These shifts are total1y consistent with those found for the equation ofcenter (about 44°, 82°, and 162° for Mars, Jupiter, and Satum, respectively). Indeed, subtracting these 260 1. Chabás and B. R Goldstein

numbers for each planct, we find 0° (Mars), about _200 (Jupiter), and +50" (Saturn), in perfcet agreement with the differences given by Ptolemy in Almagest Xill.6 bctween the northern limits on the dcferent and the apogees of each superior planet, respectively. Thus, it is quite clear that the compiler oC Vimond's tables, whether Vimond or not, had a gaod understanding ofthis difficult issue as it is presented in theAlmagesl.

Table 19: latitudc oC Mars Cf. 6v)

(1) (3) (4) (S) (6) s (0) mm min * mm min *

O 12 SO O; 9 M O; 4 O 24 55 0;13 M O; 6 1 6 59 0;16 M O; 9 1 18 59 0;21 M O; 15

4 12 8 2; 1 M 2; lO 424 S 2;34 lO 2;56

6 O S 4;21 43 7;30

10 12 S 0;21 2 O; 15 10 24 10 O; 16 [blank] O; 9 11 6 22 O; 13 M O; 6 11 18 33 O; 9 M O; 4 12 O 43 O; 6 M O; 2

* Despite the headings, these columns display degrees and minutes.

Columns 4 and 6 display the inclinatían (deciinatio minutj dial1letri) and the slaot (reflexio minuti diametn) for the inferior planets, and the entries are given in degrees and minutes, despite the headings, which read "minutes and seconds". For the superior planets, these two Ear/y Alfonsine Aslronomy in Paris: The Tables o/John Vimol/d (/320) 261

colunms display the northem and southem limits (both labeled latiludo minuti diametn) and are given in degrees and minutes.

Table 22: latitude ofJupiter (f. 7r)

(1) (3) (4) (5) (6) , (0) mm min * mm min *

O 12 10 1; 8 O 1; 6 024 12 1; 9 M 1; 7

3 6 60 1;33 M 1;33 3 18 60 1;39 M 1;39

6 O 12 2; 5 M 2; 8 6 12 O 2; 3 O 2; 6 624 S 2; O 12 2; 3

9 6 S 1;27 60 1;26 9 18 S 1;21 60 1;21

11 18 S 1; 8 34 1; 6 12 O S 1; 6 12 1; S

* Despite the headings, these columns display degrees and minutes.

The extrema1 values of columns 4 and 6 in the texl are shown below:

Mercury 4; S° (for 6s 0°) 2;29° (for 3s 18°-4s 0° and 8s 00-8s 12°) Venus 7; 12° (for 6s 0°) 2;30° (for 4s 12° and 7s 6°) Mars 4;21 0 (for 6s 0°) 7;30° (for 6s 0°) Jupiter 2; 5° (for 6s 0°) 2; 8° (for 6s 0°) Satum 3; 2° (for 6, 0°) 3; 5° (for 6, 0°) 262 1. Chabás and B. R. Goldstcin

Table 25: latitude of Saturo (f. 7v)

(1) (3) (4) (5) (6) s (0) mm min· mm min·

O 12 [blank] 2; 5 9 2; 3 O 24 2 2; 7 [blank] 2; 4 1 6 14 2; 10 S 2; 7

3 18 60 2;39 S 2;39

5 18 33 3; I S 3; 3 6 O 22 3; 2 S 3; 5 6 12 10 3; 1 S 3; 3 624 N 2;59 2 3; O

9 18 N 2;21 60 2;21

11 18 N 2; 5 33 2; 3 12 O N 2; 3 22 2; 2

• Despite the headings, tbese columns display degrees and minutes.

These extremal values in Vimond's tables agree with those in the ToJedan Tables wilh two exceptions, one of which is a trivial variant for Mercury. BU1, as faf as we know, the maximum value for the inclinatían of Venus in Vimond's tabJe is nol attested in any other previous texto It is probably significant tbat this value ¡aler appeared in Ihe edifio prillceps of the Alfonsine Tablcs (1483), as indicated in Table 13A. Early Alfonsine Astronomy in Paris: The Tables oiJohn Vimond (/320) 263

Table !3A: extrema! planetary latitudes

Almagest al-BattanT ToJedan T. Vimond Paris. Alf. T.

Mercury 4·, 50 4; 5° 4·,,50 4· 50 4·, 50 -2;30° -2;30° _2;30° _2;29 0 -2;30° Venus 6;22° 6;22° 7;24° 7;12° 7; 12° -2;30° -2;30° -2;30° -2;30° _2;30 0 Mars 4;21 ° 4;21° 4;21° 4;21° 4;2!0 -7; 70 -7; ]O -7;30° -7;30° _7;30 0 Jupiter 2; 4° 2; 4° 2·, 50 2·, 50 2·, 80 -2; 80 -2; 80 -2; 80 -2; 80 -2; 80 Saturo 3·, 20 3·,,20 3· 20 3; 2° 3·, 30 -3; 50 ~3; 50 -3; 50 -3; 50 -3; 50

For the inferior planets, between columns 2 and 3 and between columns 5 and 6 we are also given some indications ("North" and "South") to help the user. Column 7 appears only in Table J3 (Mercury), and it seems to be outside the general framework of the tableo Its entries are given in seconds and result from dividing the corresponding entries in column 6 by 10. This probab!y corresponds to the instructions given by Ptolemy in Almagest XIlI.6: to compute Ihe true minutes of proportion for lhe slant, add 1/10 when the argument lies between 90° and 270°, or subtract l/lO when the argument lies between 0° and 90° or 270° and 360°. Whether tabulated or nol, these instructions are rarely found in the medieval Latin literature on the planets (Goldstein and Chabás 2004).

f. 7v Table 26: yearly radices

This table displays the radices for the mean motion (motus) and argument (argumentum) of the fixed stars for intervals of 76, 152,304, 608, 1216, and 2432 years. Vimond does not give a radix for a specific year but perhaps this infonnation was in the canons that we have not found. As we shall argue (see Table 27, below), it is likely that the epoch of this table was also 1320 or a date c10se to it, that is, the epoch is consistent with our dating of the other radices. 264 J. Chabás and B. R. Goldstein

Table 26: yearly radices (f. 7v)

s (") s (")

years 76 years 608

mean motioo O 0;33,32 mean motian O 4;28, 3 argument O 3;54,46 argument 1 1;16,21

years 152 years 1216

mean motian O 1; 7, 2 mean motian O 8;56, 4 argument O 7;49,16 argument 2 2;32,27

years 304 years 2432

mean motioo O 2;14, 3 mean motion O 17;52, 5 argumenl O 15;38,18 argument 4 5; 4,38

In 76 years the value in the text for the mean motian of the fixed stars is 0;33,32° and in 2432 years it is 17;52,5°, corresponding lo O;O,O,4,20,56°/d and O;O,O,4,20,42°/d, respectively. These values are equivalent lo 48,954 years and 48,999 years, respectively, lo complete one revolution, or 10 in about 136 years, as in the linear lerro in the standard Alfonsine model for trcpidation which is based on Qne revolution in exactly 49,000 years. These differences in the periods depend on the seconds in the entries in Vimond's table and have no astronomical significance. However, they indicate that Vimond is not using the standard table for mean motion of the apogees and the fixed stars in the Parisian Alfonsine Tables (Ratdolt 1483, f. d4v). ln 76 years the value in the text for the mean motion of the argument for the fixed stars is 3;54,46° and in 2432 years it is 4s 5;4,38°, corresponding to O;O,O,30,26,47°/d and O;O,O,30,24,52°/d, respecrively. These values are equivalent to 6,992 years and 7,000 years, respectively, to complete one revolution. The periodic tenn in the standard Alfonsine model for trepidation is based 00 ane revolution in exactly 7,000 years, &u/y Alfonsine Astronomy in Paris: 7ñe Tabfes alJohn Y¡mand (/320) 265

and it corresponds to 0;0,0,30,24,49°/d. These differences have no astronomical significance, but indicate tha... once again, Vimond is not using the standard table for mean motion of access and reccss in the Parisian Aifonsine Tables (Ratdolt 1483, f. d4r). In fact, an entry for the mean motion of the argument is 7 times the corresponding entry for the mean motion ofthe linear termo As in the Parisian Alfonsine Tables, Vimond separates t\Vo terms for trepidation: a linear teno which corresponds to the difference between the calendar year of 365;15 days and a fixed tropical year, and a periodic terro which corresponds to the difference between a variable sidereal year and ¡he calendar year of 365; 15 days. But in his other tables Vimond has used a fixed sidereal year: we are unable to aceount for this ineonsisteney. To be sure, Vimood's canons may have explained what he intended.

f. 7v Table 27: motion of the fixed stars

The argument is given at 6°·intervals from Os 6° to 12s O" and the equation of access and recess (here called motus) is given in degrees and rounded to minutes. In Table 27, below, the edil'ors have supplied a minus sign in a few entries, where appropriate. The table has a maximum of 17;17° for argument 204° and a minimum of -0;43° for argumeot 24°. These extremal values are 18° apart (= 17; 17° + 0;43°); hence the amplitude of the sinusoidal curve corresponding to Vimond's table is 9°. This is indeed the characteristic parameter of the table for the equation of access and recess in the Parisiao Alfonsine Tables, whose maximum is 9° for argument 90°. Comparison of the entries in both tables shows that the curve represeming Vimond's table is the same as tha! used by olher Parisiao astronomers oC his time but shifted in two ways: 247° 00 the x~axis and­ 8; 17 0 on lhe y~axis. In facr, the entries io Vimood's tab1e can be dcrived from thosc io the Parisian Alfonsine Tables by taking an argumcnt and its corrcsponding equation io Ihe lalter (where they are giveo to secoods) and then adding 113 0 to the argument and 8; 1JO to rhe equation. 266 1. Chabás and B. R. Goldstcin

Table 27: motion ofthc fixed stars (f. 7v) argumentuJIl motus argumentum motus (') (0) (') (0)

O 6 -0;19 6 6 16;53 O 12 -0;)3 6 12 17; 7 O 18 -0;41 6 18 17; 15 O 24 -0;43 6 24 17;17 1 O -0;39 7 O 17; 13 1 6 -0;29 7 6 17; 3 1 12 -0;13 7 12 16;47 1 18 O; 8 7 18 16;26 1 24 0;35 7 14 15;59 2 O 1; 6 8 O 15;28 2 6 1;43 8 6 14;51 2 12 2;24 8 12 14; 1O 2 18 3; 8 8 18 13;26 2 24 3;56 8 24 12;38 3 O 4;47 9 O 11 ;47 3 6 5;40 9 6 10;54 3 12 6;35 9 12 9;58 3 18 7;30 9 18 9; 4 3 24 8;26 9 24 8; 8 4 O 9;22 10 O 7; 11 4 6 10; 18 10 6 6; 16 4 12 11; 12 10 12 5;22 4 18 12; 4 10 18 4;30 4 24 12;54 11 24 3;40 5 O 13;41 12 O 2;53 5 6 14;24 11 6 2; 10 5 12 15; 4 11 12 1;30 5 18 15;39 11 18 0;55 5 24 16; 15 11 24 O; 19 6 O 16;34 12 O O; O Earfy A/[Ollsílle AslrollOmy in París: The Tables 01John Vimond (J 320) 267

Vimond's tabJe begins at a point that in the Parisian Alfonsine Tables corrcsponds to a value of the equation oC -8;17" and an argument of about 247". The value for the equation ofaccess and recess that Vimond thought correet for his time was 8; 17", and he shiftcd the curve (i.e., the entries in the table) accordingly; indeed, caleulation of the periodic tenn in lrepidation with the paramelers for 1320 in the Parisian Alfonsine Tables yields 8; 17" exactly:

1320 . 0;3,5,8,34, 17°/y ~ 67;53° radix Incamation 359;13

Total 67; 6 and

Note that 67;6" + 180" = 247;6" or about 247", and 360" - 247" = 113" which is the phase angle of lhe shift introduced by Vimond. This table establisbes a strong connection bctwcen Vimond and the Parisian Alfonsine Tables, for this theory oftrepidation is not found in any previous text. But again, since lhe mean morions are different (see Table 26), we see no reason to assume that Vimond based his theory 00 the Parisian Alfonsine Tab!es. Rather, Vimond may have depended on an Andalusian or Castilian tradition that \Vas closely related to (but distinct from) the Castilian Alfonsine Tables, for there is no hint ofphase shifts in the Castilian canons.

f. 8r-v Table 28: fixed stars

This table displays the longitude, the latitude, and Ihe magnitude of 225 stars and nebulae but, in general, their names are omitted. The list is too long to be related to an astronomieal instrument, and the absence of star names makes us wonder what purpose it was intended to serve. Both coordinales are given to minutes. The stars are divided into three groups, in turo divided into severa! subgroups according lo the associaled planets, a feature which is certainly not common. Group 1has 137 stars that belong to lhe zodiacal conslel1ations arranged in 52 subgroups, group II has 44 stars in northern conslellations (19 subgroups), and group III has 44 slars 268 J. Chabás and B. R. Goldslcin

in southem constellations (19 subgroups); the total number ofsubgroups is thus 90. \Ve note the balanced representatíon of the stars 00 both sides of (he zodiaco \Ve have faund the same table in an early 14th-century copy: Cambridge, Gonville and Caius Collcge, MS 1411191, pp. 377-382 (rar an excerpt, see F. S. Pedersen 2002, pp. 1507-1508). There are sorne cases where an entry in one copy does not agree with the value in, or derived from, Ptolerny's treatises in contrast to the other copy, bUl there are al so examples where entries in both copies do 001 agree with those in Ptolemy. 00 the other hand, in all cases where thefe is a blank entTy in one copy, it is filled in the olher copy. In the Paris copy ooly 18 star names are given whereas in the Cambridge copy this number is reduced to 15. The star llames in these copies are generally not identical, and they are not always ascribed to the same stars. For instance, the names "almalak" and "almalac" are attributed, respectively, to the star in the 20th subgroup (MS Paris) and to the first slar of the 8th subgroup (MS Cambridge). The star list does not bear a general títle in the París copy bUI the Cambridge copy reads tabula de disposirionibus slellarum JlXarum existentibus ad terminum complementi radicis mediarllln coniu/lctionum solis el lunae quae alibi signantur. Et primo de dispositionibus illarum stellarum quae Sllnt prope viam solis. (Here begins the table on Ihe groups of the fixed stars as they were at the point of completion [the epoch?] of the radix of the mean conjunctions of the Sun and the Moon specified elscwhere. First come the groups ofthose stars close to the zodiac [lit.: the path ofthe Sun].) The first sentence serves as a general title for the table, and the second sentence is a heading for the groups in the zodiacal constellations, corresponding lO the headings in both manuscripts for the groups in the Ilorthem and southem constellations. The expression "Ihe radix of the mean conjunctions" seems to refer lO the radix given on r. Ir, "13;54,54d", which we identified with March lO, 1320. But we do not understand the expression "at the end of the complement". Madrid, Biblioteca Nacional, MS 4238, ff. 65v-66v, reproduces the same star list except that the signs used here are of60°, contrary to the other manuscripts containing this lisl. We are grateful to Paul Kunitzsch for infonnation on two additional copies of the same star list: Erfurt, Universitatsbibliothek, MS Amplon.2°395, fr. 104v-105v; and Munich, Bayerische Staatsbibliothek, MS Clm 26667, ff. 46v-47v (cf. Kunitzsch 1986a, p. 96, n. 10, and p. 98, n. 44). In both manuscripts Ihe list is anonyrnous, but in the Erfurt copy a Early A/fonsine Aslronomy i/l Paris: 1l/e Tabfes 01JohlJ Vimond (/320) 269

marginal note (in the same hand as the list) reads: Notandum istas stellarum rabulas jllisse equatas ad annum domini J338 (f. 105v). As Kunitzsch suggested 10 us (in a private communicalion), this marginal note may have been added by the copyist and not belong to the original list; no date appears in the other three manuscripts. In fact, the list in the Erfurt MS has two extra stars: one is added 10 the northern constellations, in subgroup 7 (Bootes), and the other to the southern constellations, in subgroup 6 (Eridanus). We also note thal in the list for the southern constellations the stars in subgroup 19 (Ara) are located in the Erfurt MS between subgroups 4 and 5 in the manuscripts in Paris and Cambridge (we have nol seen the manuscript in Munich). Another special feature of the Erfurt MS is that the subgroups are not numbercd; ralher, most are given the name of a slar belonging to them or even a generic name. But its main distinguishing characteristic is that the subgroups have no associated planets, in contrast to the copies in Paris and Cambridge. Hmay be of interest that the 5 manuscripts of which we are aware that contain this star list are spread all over Europe: 2 in Gennany, 1 each in England, France, and Spain. The order and the grouping of the stars in this 1ist is peculiar, for they do not follow the paltern of the catalogue in Ptolemy's Almagest that was generally adopted in medieval star lists and catalogues. Rather, this list is organized according to Ptolemy's Tetrabiblos, a handbook on astrology written by Ptolemy after the Almagest. It was translated severa1 times (rom Arabic into Latin: in 1138 by Plato of Tivoli, in 1206 anonymously, and in 1256 via Castilian at the court of Alfonso X by Egidius de Tebaldis (Chabás and Goldstein 2003a, p. 232), and was known as the Quadripartitum. In Tetrabiblos 1.9, Ptolemy grouped the stars into lbree main categories (zodiacal, northern, and southem constellations), following an order differing from that in the Almagest where the northern constellations precede the zodiacal constellations, and grouped the stars within each category according lo their associated planets. As an example, we reproduce a passage of Tetrabiblos 1.9 eorresponding to the stars in the eonstel1ation of Aries (Robbins 1940, p. 47):

The stars in lhe head of Aries, then, have an effeet like the power of Mars and' Saturn, mingled; those in the mouth like Mercury's power and moderately 1ike Satum's; those in the hind foot like that ofMars, and those in the taillike thal ofVenus. 270 J. Chabás and B. R. Goldstein

As is readily seco, the arder, the subgroups, and the planets associated with the stars in Aries in Vimond's list perfectly match ¡hose in Ptolemy's Tetrabiblos. And this is indeed the case for almos! all stars in the 90 subgroups displayed in Vimond's lisI. The star positions generally agree with those in Gerard of Cremona's version of Ptolemy's slar catalogue in the A/magest with an ¡ncremen! in longitude of 17;52° for precession, a value otherwise unattested. Ifthe rate ofprecession was taken to be l° in 66 years, 17;52° would correspond to about 1179 years and, if we add it to 137 A.D. (the date of the star catalogue in the Almagest), we gel 1316 A.D. Bul it is nol clear that this date had any significance for the author. We have compared this list to that in the Libro de las estrellas de la ochaua espera (Madrid, Universidad Complutense, MS 156; see aloo Rico Sinobas 1863-1867, vol. 1, pp. 5-145), also known as Libro de las XLVII/figuras de la VIII spera or even as Libro de las estrellas fixas. This is an adaptation of the star catalogue for 964 AD by the Persian astronomer al-$üfi (903-986) which in tum depended on the star catalogue in Ptolemy's Almagest (see Comes 1990). This work, where the total precession is 17;8°, was compiled in 1256 by Judah ben Moses ha-Cohen, one of most distinguished collaborators of Alfonso X. The presentation of the star data in this Alfonsine text differs substantially from that of a typical star Iist although the data themselves are what one would expect, namely, for each star we are given its name, longitude, latitude, and magnitude. The associated planets are aloo given for each star, afien adding an indication of their relative strength, showing that the Alfonsine Libro ultimately relied on Ptolemy's Quadripartitum. Howevcr, after comparing the data in the Libro with those of Vimond, we see no evidence to suggest that the star list found among Vimond's tables is systematically related to this Alfonsine book. As Kunitzsch infonned us, there is a star list by John of Ligneres containing data for 276 sl'ars, but the longitudes are Alfonsine, i.e., Ptolemy's values plus 17;8°: BibliothCque nationale de France, MS lat. 10264, ff. 36v-38v, and Florence, Biblioteca Nazionale Centrale, MS Conv. soppr. J.4.20, fols. 214v-216r. This list was extracted from the star table that later appeared in the edilio princeps of the Alfonsine Tables (1483), and sheds no additional Iight on the list included in Vimond's tables. Moreover, in the course of examining the star names in the four manuscripts containing this list, Kunitzsch noticed that the author drew upon a variety of Latin sources, mainly the translations of the Tetrabiblos Early Alfonsine Astronomy in Paris: The Tables o[Jollll VimOlld (1320) 271

but also sources oot in the Telrabiblos lradition (sorne of which cannot be identified). Thus, Vimond's list is dependent on Ptolemy in two ways: the choice of the stars, their arder and grouping, as well as the associated planets, are borrowed from the Quadripartitum; and the numerical data are taken from the Latin version oftheAlmagest. In sum, we believe that the star list attributed to Vimond in the Paris MS, and that is anonyrnous in the Cambridge, Erfurt, Madrid, and Munich MSS, derives from an unknown archetype; we know ofno similar star ¡ist in Latin in the 14th century or in the previous Arabic literature with which to compare it. In Table 28 \Ve present in the first 3 colunms a complete transcription of the Paris copy with translations of the headings and the names of the associated planets in each case. For the latitudes ''north'' is indicated by an abbreviation ofthe tenn septentrionalis, and "south" by an abbreviation of meridiollalis; we have replaced them with the modern designations + and -. Column rv gives the few star names found in the Paris copy, which \Vere added in interstitial spaces within the table (sorne of the star names are partly hidden in the gutter of the manuscript and cannot be read compietely); column V lists the modem star designation; column VI gives the standard number assigned to each ofthe 1028 stars in Ptolemy's catalogue; column Vil offers comparisons and cornments, together with variants in the Cambridge copy; and column VIII provides the identification of the star names. Table 28: star list (r. 8r-v) N~ N

(Constellation] Associated planets

, 11 III IV V V, VII VIII

Longitude Latitude Magn. N""" Modem Number Comparisons ldenlifieation of designation (P.-K.) and commenls Star names (sign) (degrees) (degrees) ,. Zodiacal eonstellations] I [Aries] Mars, Saturn ~ O 24;32 + 7;20 3 Ari 362 [ O 25;32 + 8:20 3 PAri 363 l'" 2 [Aries} Mercury, Salum f,' O 28;52 + 7;40 5 fI Ari C(IlI): blank 36' ~ C(IV): flamai? Unidentificd 5 O 29;22 + 6; O 5 Ari 365 C(1lI: blank Unidcnlificd: C(IV): hercules sec 11Gem, below

3 (Aries] Mm , 2;52 p2.J , Col 374 C: +$;1$, 0:-$;1$ , 5;52 ~ 5 Ari 373 C: +1;30 , 7;32 - 1;20 5 ) Ari l72 C:+1:20 Ir., ':'0 +1'10 4 (AriesJ Venus 1 9;12 4;50 , Ari l68 1 11;42 + 1;40 4 Ari l69 ...~ 1 13;12 + 2;30 4 Ari 370 ~ 'i;; 1 14;52 + 1;50 4 Ari J71

~. 5 LTaurus] Venus.~ C: Moon •~ 1 17;32 9;30 5 30(e) TlIu l84 1 21;32 8- O l ATau l85 C:+8; O I ~ 6 [Taurus. The Pleiades] Moon, Mars ,- ~ 1 20; 2 4;30 5 19Tau 409 R' 1 20;22 4;40 , 23 Tau 410 l 1 20;32 + 5; 5 5 27 Tnu' 412 ~ 1 21;32 +5;20 5 useIISS' 411 ¡¡-'" -a. 7 rraurus] Mars :;. ir 2 I 0;32 5;10 1 aldebaran? Tau 19l C(IV): aldebaran lo,p. 89 n. 10, cte. 8 fTaurusJ Saturn, Mercury [ 1 26;52 5;45 l T" 390 C(IV): almalac Ifthis is ti corruption of ." IArabic al·mal/Ir. (Ihe w~ ing), ít should designate InLeo (Regulus), See O, '" p. 101 n. 12. ~ '"w N 1 8;42 ] I Tau ]92 c: +S;SO i=-">º G: -SiSO, -0;50 "i 1 29;42 b..L.2 ] Too ]94 :+3; O 2 3;32 foi;....2 4 Too ]99 :+4; O . G: -4; 0, +4: O 9 [Tauros) M"" 2 7;52 3;30 , 100(1)Tll.u ]97 2 8:12 ,. O , \Q4(m) Tau 396 C: +5; O

2 13;]2 + 5; O ;. f3Tau 230/400 G, 3

2 ~2 2;30 3 Too 398 G+ 17;52: 15; 2,15;32 ~ 10lGemini] Mercury, Venus [ P 2 24;22 ::..U..Q 4 'lOem 417 C:+1;30 1" 2 26; 2 1·15 4 Gom 438 C: +1;15 ~

2 28; 2 =1;)0 4 vOcm 439 C: +3:30 ~ • 2 29;52 no ] Gom 440 C: +7;30 3 2;32 1!Ul! 4 Gom 441 C: +10;]0

11 (Gemini] Satum 3 I 9;32 5;30 I 3 Gom 4lS 12 (Gemini] M"" 3 11;12 9;40 I 2 ( )annaj? Gom 424 13lGerniní] Mars

3 1432 (..J6;15 l··] hcrcules? P Gcm 42' C(1I): +6;1~.C(1lI): 2 R, p. 48: Hcraklcs C(IV): almucrcdan KI9~9,p.127:E Viris 2', alled almuredin ... 14 (Canecr) Mercury, Mars ~ 3 20;32 1; O , C" 456 '" l· 3 25; 2 7;30 4 C" 4S7 ~

15 (CancerJ Satum, Mereury ~

3 26;12 11;50 4 I Cne 4SS ~ S· 4 4;22 5"30 4 la Cne 454 G: -~;30 16 [Caneer] Moon, Mars ~

3 28;12 0;40 n rneollef7 OC 2632 449 C: 2. C(IV): mellef7 P, r. I~va:meelef ~ Oalaxy M 44 .. 17 [ClIneer] Mars, Sun ..,~ 3 28;12 +2;40 4 assinis? rCne 4S2 C(IV): asini G .., 3 29;12 +..Q;JO 4 10ene 4S3 0:-0;10 18 [Leo] SalUm, Mars [ 4 12; 2 9;30 3 Lo. 465 ~ 4 ~ +12; O 3 k.tleo 464 G + 17;~2:12;12 '"~ 19 [leo] Satum, Mars C:Mcrcury '" 4 18; 2 11; O 3 Lo. 466 '"~ I ~ N 4 18;32 4;30 3 L,. 468 ~ '" 4 20; 2 lUJo 2 L,. 67 G; +8;30 20 ¡Leo] Man, Jupitcr

4 20;22 0;10 I lalmalak? L,. 469 p.p. 101 n. 12.

21 (Leo} Venus. Satum 4 29;12 IlJ5 5 6O(b) Leo 480 C,G:+12;15 5 2; 2 f+1];40 2 Loo 481 ,..

5 2;12 f+ll;30 5 81 Leo· 482 &> ~ 5 4;12 9;40 3 L'. 483 •• [ 5 12;22 1120 I ?? 13Leo 488 G: +11;50 !" 22lLeo] Venus, Mercury ?'

5 8;12 5;50 3 I Leo 484 5' ;¡ 5 8;22 3; O 5 L,. 487 "S'

5 9;32 ~0;50 4 L,. 486 5 9;32 1;15 4 L,. 485 23 [Virgo] Mercury, Mars

5 14;12 4;35 5 " Vir 497 5 14;52 " 5;40 5 Vir 498

5 11.;...1 6; O 3 IDVir 501 G + 17;52: 16;52 5 18: 2 5;30 5 I Vir 500 24 [Virgo 1 Mercury, Venus

5 26; 7 1;10 3 Vir 502 ~ ~ 6 1; 2 1<-2;50 3 Vir 503 ..,>. 25lVirgo] Saturn, Mercury l· I 6 O; 2 15;10 3 I Vir 50' >. 26 [Virgo] Venus. Mercury I 6 14;32 2; O 1 lalmurc? Vir SlO qIV): alcirncch G: ascimech I ~, 27 [Virgo] MercuIY,Mars ;p

6 24;32 7;30 4 I Vir 51' ~.

6 25;12 f+2;40 4 Ji: Vir >J9 ir 6 ..l;ll 0;30 4 Vir 521 e, G + 17;52: 27;52 ~.. ~ 7 0;32 9;50 4 Vir 522 ~ .. 28lLibraJ Jupill:r, Mercury g- 7 5;52 0;40 2 ~Ub 52'

7 10; 2 8;30 2 ~Lib 531 [ 29 [Libra] Satum, Mercury "::; 7 9;12 [..]1;15 (..( IvLib 534 C(JI): +1;15. C(III): 4 <>

7 11;52 (..]1;40 (..( I Lib m C(II): +2;40. C(1lJ): 4 '" " 7 15;22 [..}3;45 [.·1 Lib C(IJ): +3;45. C(IlI): 4 ~t:l 7 0;52 [..J4;30 [..1 Lib '"53. C(II): +4;30. C(1lI): 4 JO(Scorpiusj Mars,Salum 7 23;32 1;40 3 S,. 5" 7 3;32 b..LlI 3 S,. 548 C:+5; O

7 24;12 ~ 1;20 3 13Seo 54. 31 (Scorpiusl Mars,Jupiter ,. I 8 i.= 4; O 2 S,. m G + 17;52:0;32 32 (Scorpius] Salurn, Venus ~ e:- 1+ 1S 8 15; O 4 1l CO 558 O + 17;52: 6;42 W. ~ 8 11; 2 12;1Q 3 1] Seo 5.' C: +19;30 ,., ?' 8 16; 2 18'~ 3 S,. 5.2 C: +18;50 ., + 17¡52: 16;52 8 3 " Seo 5" G o: iWl f!llº C(lI): +16;10 ~. I 8 18¡22 1M2 3 ,1Seo 5.3 C: +16;40 • 33 [Scorpiusl Mcrcury, Mars,M.22n C: [blankl 8 F=l 4 S,. 5•• e, G+ 17;52: 14;52 = C. 0:-13;30 ¡H;ll Fillº s,. 5.5 C,G+ 17;52: 15;22 8 3 C(lI): +13;20

J4¡Scorpius) Mars.Moon 8 19; 2 f-13;15 n G Seo· 167 C: 2 COlo 6441 35 rSaginariusJ Satum, Moon i:' ~ 8 22;22 6;30 3 SO' "O ~ ';;; 9 0;52 =.1áQ 4 SO' "6 C: +3;50 ~. 36 [Sagiuariusl Jupilcr, Mars

8 24;32 2; 7 4 ~ Sgr S74 f ~ 8 26;52 1;30 3 AS", m s· ;.­

37 ¡Sagittarius] Mercury, l.1!nilkLSun, Mars C: Moan ~,

9 3; 2 =.1& n vI + v' Sgr m G: -0;45. qIV): 2 ~ ~.. 38 [SagittariusJ Jupitcr, Mcrcury a­ ~ 9 4;12 6;45 3 SO' '9' :. •~ 9 5;32 =UQ 4 S", '90 C: +2;30, G: -4;30 i 9 7;52 =UQ , \ji Sgr 589 C; +2;30 i ." 39 [Sagiuarius] Jupiter, Satum ::: ." 9 4;52 18; o 2 SO' '93 ;¡¡'" 9 5;32 ~ , pI +p2Sgr C: +23; O ~'" ,,,59' o 9 ll;ll =lt...Q 3 " Sgr G + 17;52: 24;]2 C(ll); +13; O 40 [Sllgittarius] Venus, Satum

9 ~ 5;50 , 59(b) Sgr ,,, C,O+ 17;52: 16:22 9 ~ ~ , 60(A) S", 59' G+ 17;52: 1532 CnJ): +4;50 ,. 9 16;42 ~ , c: +4;50 " Sg, '" Q 9 17;32 =.JUQ , 62(c) Sgr 600 c: +6;30 ~ 41 (Cllpricornus] Murs, Venus [ ", 25;12 2;20 3 1+cr.2Cap 601 9 .,1" 9 25;12 5; O 3 C'P 603 f. 9 26;42 1;30 6 C'P 607 • 9 6;52 0;45 6 C'P 60'

42 [Capricornus) Mars, Mcrcury

9 29;32 8;40 4 (O Cap 612

10 4;32 =.MQ 4 24(A) Cap 613 c: +7;40 ,., 10 , - =..Q;>Q 4 C'P 614 c: +6;50 10 I 8;12 .§.;...O 5 36(b) Cap 615 C:+6: O 43 [Capricomus] Satum, Mc:rcury

10 12;42 ~ 3 e.p 623 G: +2;10, -2;10 ~ -<" 10 14;12 2-º 3 e., 624 G: +2; 0, -2; o >. 10 14;42 ~O 4 2(d) Cap 62> G: +0;20 "" l· 10 15;32 2~ 5 e., 627 e, G: +2:50 >. 44 [Aquarius] Satum, Mercury I 10 2;32 8;40 3 Aq' 636 ~ S· 10 4; 2 8; o 4 Aq' 6J5

10 14;22 8;50 2 Aq' 6J2 ~ :j 10 24;12 f+-JI:15 1 Aq' 6JO G: +11;0. G: 3 , <;J 45 [Aquarius] Mercury, Saturn ~ ~ 10 J..2.;il 5; o 4 Aq, 647 0+17;52:29;12 .., 10 ~ f..U!! 3 Aq' 646 0+ 17;52: 29;32 .. e,o:+7;30 .. 10 2;32 ~O 5 53(1) Aqr 648 c: +5;40 " 46 [Aquarius] Satum, Jupiter l 11 5;32 1; o 4 83(h) Aqr 653 '"::: 11 6;52 F-Uº 4 \VI Aqr 656 C: +7;30, G: -8;30 '" 11 7¡52 I=-0·JO 4 cpAqr 654 C: +7;30 ~N N 1\ 8;12 .1.;10 4 a[n]phora Aq' 65l C: +1;40 See note l. 00 I N 47 [pisces) Mercury, Satum 11 9;32 9;15 4 p" 674 1\ 12; 2 7;30 4 p" 675

11 13;52 9;20 4 7(b) Psc 676 48 {pisces] Jupitcr, Mercury

11 13;52 4;30 4 " Psc 679 ,.. 11 17;32 2;30 4 ¡.p" 680 9 49 (Pisces] Satum, Mercury ~ [ 11 23;52 6;20 4 (o) Psc 681

11 28;52 5;45 6 41(d) Psc 682 "?'

SO[Pisces] Jupiter, Venus ~ ¡¡ 17-12 J7;52: J7;22 O 15;20 4 p" 706 a + 5 O 20; 2 t+17; o 4 p" 70S SI [Pisces] Satum, Jupiler

O 13;32 14;20 4 11'1Psc 702

O 14;12 13; o 4 \jI1 Psc 703 O 15;32 12; o 4 p,,' 704 52 [pisces] Mars, Mercury 1 O 120;22 8;30 I 3 1 ~ p", 692

~ '<" ~ [Tille:] Then follow ¡he conslellations (dispositio) ofthe other fixed stars in {he nonhcm pan. S; (Nonhcrn constellalions: al11atirudcs are posilive] ,.~ I [Ursa Minor] Saturn, Venus "~

4 S; 2 72;50 2 alicdin UMi 6 C(IV): alicdim Scc nolc 2. 4 14; 2 74;50 2 alforcami Umi 7 C(IV): alfoza K1961, p. 58: al-farqadan j (I3+YUMi) s· 2 [Ursa Maior] Moon, Venus ~ , O; 2 2 UM. 33 G: 13;30 iU!l ;;! , 5;52 2 benczna Um, 34 G: 15;40 K1966, p. 42, no. 23: ~ ;¡<" benenaz (11UMa) ~

, 17;42 :!>....O 2 UM, 3l G: 14; O .., •.. 3 [Draco] Saturo, Mars ;¡- , 26;22 84;50 3 De. 67

, 27;52 8..8.;...0 3 De. 68 G: 78; O [ c:, 4 [Cepheus] Saturn, Jupiter ::: O 4;32 69; O 3 Cep 78 ~~ '"

4 13 77 O 25;22 1;10 Cep ~N w , ~ 11 27;12 72; O 11 79 I Cep ~ S [Hercules] Saturn, Mars 7 28; 2 53:30 , Her 130 O: 13;30 7 lMl ~ 3 ¡c;Her 129 0+17:52:21;42 G: 16;10 8 1;52 ~ 3 InHer 133 G: 19;50. 59;50 8 3;12 60;20 , 69(e) Her 13' ,.. 61Corona Borealisl Venus, Mercury 9 6 29;32 46;10 , P GrO 112 ~ 7 2;32 44;30 2 alreea ~C,B 111 C(IV): aireen G ~ 7 S; 2 44;45 , yerO liS !"

7 7; 2 44;50 , C,B 116 el "o ¡¡: 7 [Boolcs] Mercury ,.;¡ 6 I 9;12 28; O I 3 I "Boo 107 8 [Lyra] Venus. Mercury

1 1 1 9 [perseus] Saturn, Jupittr

1 1J7;29 23; O I 2 leiumezuz? ~ Pcr 202 G+ 17;52: 17;32 Unidenlified 10 [PerseusJ Mars, Mercury 1 122;42 30; O I 2 I laPcr 197 II [Auriga] Mars, Mercury I 2 12;52 22;30 1 alhaioch laAur 222 C(IV): alhaioch G 12 (Ophiuchus] Satuffl, Venus ...'"~ 8 12;42 36; O 3 alhanue a Oph 234 C(JY): alhanue K1966, p. 55, no. 33: '­ ';;; alhaue, alhane

13 (Serpens) Satuffl, Mars i·o '- 7 12;12 25:30 3 laSer 271 G: 25;20

7 12;42 36:30 i..Ser 270 C, G: 26;30 • .;;i 7 14;12 24; O J f; Ser 272 s·

7 16;32 16;30 ¡.l Ser 27J • ~ 14 [Sagitta] Mars, Venus

9 24;32 39;10 ¡; Sge 282 " 6 ~" 9 28; 2 39;20 YSge 281 ~'" • .g, 15 (Aquilal Jupiter, Mars :;- •~ 9 21;42 29:10 2 vultur Aql 288 C: 19;10 G C(lY): vultur ~ ~ 16 [Delphinus] Satum. Mars ~ 10 6;22 32; O J PDel JO' '"N 10 8; 2 33;50 J D,I JOS '" N 10 9;12 32; O J .sDel J06 00~ 10 111;22 ll.;lQ 3 I ,Del 307 G: ]];10 ~'" 17 (Pegasus] Mars,. Mercury '" O 1>'-.1 12:31 po. 316 G + 17;.52: ID: 2 • G: 12;30. C. G: 2 1I 20; 2 31; O 2 PPcg JI7 18 [Andromeda] Mars, Venus

O 13;32 15; 7 3 TJAnd 341 ,.. O 19;42 30: O 3 Aod 347 9 O 19;52 32;30 3 vAnd 348 ~ O 25:42 26:20 3 pAnd 346 a + 17;52: 21;42 C(ll): +16;20 [ I 4;42 23; 3 yAnd 349 " o " 19lTriangulum] Mcrcury ~ O 28;52 16;30 3 Tri ll& ~ s· I 3;52 20;40 3 Il Tri JS9

(Tille:) Then follow (he constellations (dispositio) orthe other Sl8rs in lhe southem parto (Soulhem constellations: alllatiludes are negative)

1 [Piscis AustrinusJ Mars. Venus, Mercury I 10 I 9;42 16;30 I 4 I p", 1020 10 J.§.J 15; O 4 " PsA 1019 C,G+ 17;52: 13; 2 10 16;42 il.;...Q 4 "PsA 10J8 G: 14;40 10 18;32 20;20 4 Il PsA 1012 ,2' ... 2 [CetusJ Saturn ~ O 112;52 20; O 2 I l;;Ce! 725 " l· 3 [Orian] Mars, Mercu'Y ~

2 1,9;52 ¡7i O 1 1 a Ori 735 ~ 4 [Orian] Jupiler, Satum ~,. 2 7;42 31;30 1 POri 768

2 13;12 24;10 2 bOrí 759 ~ ;;! 2 15;12 24:50 2 t Ori 760 • ;;' 2 16; 2 25;40 2 Ori 761 ~ 5 [Eridanus] Jupiter

O 118; 2 eEri 805 CC!): 16; 2, G: 13:30 §: llJ.Q 1 1 ""• 6 [Eridanus] [.·1 C: Saturo

2 1_5;12 31;50 4 1 AEri 772 0+17;52:6:12 ."l 7 [Lepus] Saturn, Mars ~ N 2 12;42 44;20 3 P Lep 813 '" 2 13;22 41;30 3 a Lep 812 ...,'"00 N 8 [Canis Maior] Venus 00 00 2 13;52 57;40 2 a Col 845 2 16;52 lMQ 2 P Col 844 C: 57;40 9 [Canis Maior] Jupiter,Mars 3 5;32 39;10 1 aCMa 818 3 7;32 35; O , e CMa 819 10 [CanisMinor] Mercury, Mars ,.. 3 13:22 14; o , PCMi 847 G + 17;52: 12;52 Q ~ 3 17; 2 16;10 1 aCMi 848 .­• [ 11 [Hydra] Saturn,Jupiter , 17;52 20;30 2 (l Hya 905 '" Cl" o , 23;52 26;30 , " Hya 90. a: ¡¡ , 26;32 26; o , \} Hya 907 , 12 [Crater] Venus, Mercul)' , 17;52 18; O , oCn 923 , 17:52 18;30 4 Cn 92' G + 17;52: 24;52 , 20;22 19:30 , ,Cn 922

13 [Corvus] Saturn, Mercul)' • 2;12 19;40 I 3 ,C~ 929 I 6 6:22 14;50 3 YCrv 931 c: 6;12 14 [ArgoJ Satum, Jupitcr

I 3 5; 2 69: O 1 laCar 892 O: 29; O ~ 15lCentaurus] Man, Venus ~ ""-;;; 6 24; 2 25;40 3 \ Cen 939 ~. o 7 3;32 2230 ¿ C" 940 C:2 ~ 16lCenlaurus] Venus, Jupilcr 6 26:13 41;10 1 lo.Ccn 969 G+ 17;52:26;12 j C: 26;12 6 27;52 51;10 2 ,Cru 96' " ?¡;;. 6 29; 2 55;20 2 Cru 968 •;;! 7 3;12 51;40 2 PCru 966 .. 7 12; 2 45:20 2 P Cen 970 C,45;1..1 ~ .., 17 [Lupus] Venus, Mars :;- ~ 7 13;42 29;10 3 Lup· 973 • 7 15;52 24; 10 3 P Lup 972 G: 24;50 [ 181Corona Auslralisj Saturn, Mercury ~ w~ 9 4;22 U2Q 4 r erA 100' G: 15;10 9 4;42 16; O 4 C,A 1004 '" N~ 9 lli 17;10 4 13Cr A 1003 G + 17;52: 4;52 '" 290 J. Chabás and B. R. Goldstein

Col. 1, The number of the zodiacal sigo is nol repcated in col. VII whcre variants are Usted; in aU cases reportcd in that column only ¡he dcgrces and minutes differed (rom the cnlry in ¡he París manuscript.

Col. 111: n meaos nchulou$.

Col. IV; In the manuscript the names of the stars are 001 presentcd in a colurnn.

Col. V: The cotries in this column have becn takcn from Toomcr 1984. • indicates ¡hal Kunitzseh 1986 and Kunitzseh 1991, pp. 187­ 200, give a difTerenl modem designalion.

Col. VI: These numbcrs are taken from Peters and Knobcl 1915 ~ (ultimately (rom BaBy 1843), and Ihey are also used in 0 Kunitzseh 1986 and 1990.

Col. VII: e refers lo Cambridge, Gonville and Caius Collegc, ~ ~ ~ ~ ~ MS 1411191; in certain cases. il is followed by a column number ~ ~ g: in Roman numerals. G refers lO Gerard of Cremona's version 01 Ptolerny's star catalogue (Kunitz:seh 1990). We underline cntries in Vimond's lable for which there is a varlant reading. The entries for longitudes in bolh copies generally agree with those in < ~ G with an ¡ncrement of 17;52° for precession; (hose cases where • <• Ihey differ have been noled. w = Col. VIlI: G refers to Gcrard ofCremona's version of Plolcmy's slar calalogue (Kunitzsch 1990); K1959 refers to KunilZSCh , 1959; KI966 refers to Kunitzsch 1966; P refers to Plalo of Tivoli's Lalin version ofthe Tetrabiblos(ed. 1493); and R refers '"w • lO Robbins 1940. " ~ ~ ~ oc.•" Nole l. We are informed by Kuni1ZSCh that atlphQra is nol a .. proper name but rather a noun used in the deseriplion of the slar's posilion: "where (he waler flows out ¡rom fhe vcsse"; o o o- Erfur!, Universillitsbibliolhek, Amplon. 2°395, f. I05r, itl 6" ~ decursu aqlle n. ab onpnora): f. /n aque vero decursll ~ '"~ ~ P, 16va: " col/acate (withoUI ollphora).

N N N ~ Note 2. As Kunitzsch informed us, aliedim, apparcntly renders " . ~ ¡.¡ " ~ Ihe Arabic a/-jady (lhe kid), an old Arabic name for a UMi .. - - (Kunitzsch 1961, p. 62). 11 is uncertain where Ihe compiler oflhis ~ ~ lisl mighl have found iL In the Tefrabib/os lradilion this name - ~ ~ ~ never OCCUT$. Ear/y Alfolls;ne AslrOllomY;/1 Par/s: rhe rables o[JOhl1 V;/IIond (/320) 291

Acknowledgmcnts

We thank Paul Kunitzsch and Beatriz Porres for assistance with the Latin texts cited in this article, and Fritz S. Pedersen, John O. North, and Julio Samsó for detailed comments on a preliminary versíon of this papero

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